Category: mathematics

Imagine this: On the news this morning you hear a segment about the weather – the maximum temperature tonight is predicted to be -2℃ and this is 10℃ colder than last night!

Question: What was the maximum temperature last night?

Now imagine this: You step into an elevator in a tall building and travel down 10 floors to the car park. When the doors open, you see a sign on the wall saying ‘Level -2’.

Question: If the ground floor was Level 0 , on which level of the building did you enter the elevator?

Next, imagine this: You go to the cinema and spend £10 on a ticket. Later, you check your bank balance and see that it is -£2.

Question: What was your balance before you purchased the ticket?

Finally, imagine this: You sit down in your mathematics class and see the equation 𝑥-10=-2

Question: What is the value of 𝑥?

What do these scenarios have in common?

Perhaps you noticed that they all have the same numerical answer? Maybe you recognised that they are all asking the same numerical question: Which number is 10 more than -2?

We sometimes describe such problems as ‘isomorphic’ – they have the same underlying structure but different surface details. Those surface details often add context to otherwise abstract mathematical problems, and it is common for us as teachers and designers to try to include these contextual or ‘word problems’ in teaching materials. But for what purpose? I suspect the immediate answer that comes to mind is one based on ideas of numeracy or mathematical literacy – ‘because learners need to be able to apply their knowledge to solve problems in the world around them’. Another common answer might be simply ‘because these sorts of questions will come up in the exam’. A perhaps less common answer could be ‘because they help learners to understand a concept’.

But how many students would recognise that the problems posed earlier were in fact isomorphic? Or do they tackle each of these as isolated problems, never noticing the connection between them? Does it matter either way?

In the two problems below (adapted from those in Greer and Harel, 1998, p. 20), problem 2 is provided by the teacher to help a student who is struggling to solve problem 1.

1. In the diagram, 𝑎1=𝑎2 and 𝑏1=𝑏 Find the value of 𝑎2+𝑏1

1. You and your sister had £180 altogether. Your sister gave me half of what she had and you gave me half of what you had. How much money do you have left between you?

Here the teacher is supposedly trying to support the student by providing an isomorphic problem that they might find easier to make sense of. However, Greer and Harel reported that in this case, the learner saw no connection between the two, that is until after they had constructed a solution for problem 1 (which rendered the teachers’ attempt at support somewhat unsuccessful!). And this finding is not unique; it is commonly reported (e.g. Barniol and Zavala, 2010 ; Lin and Singh, 2011) that learners simply do not spontaneously recognise isomorphic problems – that which is an obvious analogy to the teacher (an expert) often remains invisible to the learner (a novice).

I am reminded of a time when my GCSE mathematics class were exploring linear graphs in the context of Celsius and Fahrenheit temperature scales. After some time, a student loudly protested, ‘You haven’t given us enough information – we don’t know this value!’ (referring to the freezing point of water in degrees Celsius). Surprised, I prompted them to think of their science classes, to which they responded, ‘Well in science it’s 0℃. But this isn’t science, it’s maths!’

This interaction has stayed with me for many years. I had made assumptions about the ways in which students implicitly connected their knowledge, that they would automatically recognise where and how knowledge could be transferred from one setting or context to another. But they didn’t, even with something as elementary as the freezing point of water.

So, recognising sameness in context matters – that there are universal facts and knowledge that can be applied to a problem, be it in mathematics, science, geography, or in what we sometimes call ‘the real world’ (that mysterious place outside the classroom). But what about sameness in (mathematical) structure – does it matter if students don’t recognise two problems are isomorphic if they can solve each of them correctly anyway?

Here I ask you to consider the following three problems, this time concerned with combining vectors. Try to reflect on the first mental image that comes to mind in each case; you might like to note or sketch something for one before moving to the next.

Problem A

There is a vector of 3 units to the east and another vector of 4 units to the north.

Sketch the two vectors and the vector sum.

Problem B

A car travels 3km to the east and then 4km to the north.
Sketch the total displacement vector.

Problem C

Two forces are exerted on an object. One force is of 3N to the east and another force is of 4N to the north.
Sketch the force vectors and the total force vector exerted on the object.

These problems are also isomorphic and did you notice, the context likely influenced your mental image, your ‘sense’ of the problem and the solution path you might take?

Let’s consider this further. Problem A is the most abstract; it requires us to draw upon our knowledge of mathematical conventions and terminology and procedures associated with adding vectors, and to undertake significant mental manipulation of the two mathematical objects (the individual vector arrows) to form a sum.

Problem B focuses on displacement and brings with it the benefits of intuition around sequential movement. Here it’s much easier to sketch the vector combination correctly and to recognise the effect of total displacement.

Problem C focusses on forces; it requires us to have a sense of what a force is and knowledge of conventions associated with free-body diagrams (most likely from physics classes). In this context, our intuition sometimes leads to initial misconceptions – for example, that the resultant vector will join the individual vectors end-to-end (completing a triangle).

Now, knowing that these problems are isomorphic, did/does any one help you to make (more) sense of another?

I am hoping that the answer is yes! As humans we are excellent at pattern spotting and building analogies and noticing, or imagining, similarities – it’s how we build connections and retrieve memories. Our ability to work flexibly with a mathematical concept is related to exactly this: our capacity to build and tap into that rich network of connected ideas, experiences and problems. But this doesn’t happen spontaneously – it is cultivated over time, an accumulation of experiences and prompts to compare and contrast situations. So, I wonder, how often do we give students time and guidance not to practise solving different, isolated problems but to understand a concept in different ways or contexts, and moreover, how often do we deliberately give students isomorphic problems presented in those different ways?

In the same way that we might support an individual student who is struggling to solve 𝑥-10=-2 by offering them a relatable, analogous scenario or way of thinking about the problem, why not incorporate explicit opportunities for all students to see, compare and even create their own isomorphic problems? Perhaps then they will begin to construct stronger, richer networks of knowledge, or ‘mental libraries’ of contexts and problem types that they can visit when confronted by a new problem or concept, seeking, not an answer, but a sense of the problem and possible paths to a solution.

Let’s finish with a challenge – how many problems can you create that are isomorphic to this?

Solve the equation 10𝑦=2

And how about problems isomorphic to this?

Solve the equation -10𝑦=2

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tabitha Gould*

Have you heard of esports? If not, then where have you been? Probably outdoors having fun with friends or loved ones … but I’m here to tell you that you’re missing out on the greatest genre of entertainment this side of a digital screen!

Should you be unaware, esports is essentially a computer game that is played in a competitive setting by the best players in the world – usually in front of a crowd. These games are often streamed on platforms such as Twitch or YouTube to millions of fans. Talking of millions, the best players in the world can earn those kinds of figures. Additionally, the esports scene was valued at $1.81 billion in 2024 and is set to reach $5.88 billion by 2030.

While you might have an image of people just sitting at computers messing around, there is in fact a huge amount of variation and flexibility to these games when played at the top level, and a massive infrastructure of people involved. There are the players, the coaches, the managers and even private chefs and personal trainers! Importantly teams also have analysts to track information about their own teams and also about the other teams as well. Each team plays differently, and each player plays differently. Numerous statistics are tracked during games, both in professional games and the general player base. It’s a huge amount of data to work with – and it’s what this blog will primarily focus on.

Now you might be saying, ‘Ray, how on earth are you going to link this to mathematics?’, and I’m here to tell you ‘By a very delicate thread’ – a trait of all my blogs.

But what I’d like you to consider as you read this, is the incredible number of variables that are being presented. Ones that are constantly at play, ones reactive to other choices, variables within variables … a data smorgasbord!

League of Legends

There is a vast array of games which have their competitive scenes, but for this blog I’m going to stick with one called League of Legends – due to my own knowledge, both in terms of playing (9,210 hours as of writing this) and also watching (since the very first season back in 2011).

In short, League of Legends is a multiplayer online battle arena released in 2009. In the main game mode two teams of five battle against each other to destroy their opponent’s ‘Nexus’ (their base). On average a game takes 25–40 minutes.

It’d be easy to think that this is just a simple 5v5 game, so you just jump into a game and get killing. But the game starts before … well, before the game starts.

The game and its variables

Before the game clock can even reach 00:01, champion selection must take place, a process which takes around 5 minutes by itself. This is where each player decides which champion they are playing in the game – a very collaborative process between all the players of the team, and the coaches. The aim is to have a well-rounded team composition built of champions that each offer something different.

In the below sections, variables 1–4 will cover this phase, while 5–7 cover aspects found during actual game play.

While we progress through these variables, I’ll try to inform you of numerical variations that are found within each one. Keep these in mind as you progress through to try and grasp the scale of knowledge required, and the breadth of data analytics required to strategise against your opponents.

Variable 1: The champions

There are currently 171 unique champions to choose from, and each one has their own abilities; each champion has four abilities as standard, though a select few have a couple more or a couple less. Each champion then has their own values for stats which encompass things such as health, attack speed, armour, magic resist, attack range, etc. Additionally, each champion has a unique passive effect that provides a continued effect throughout the game.For instance, a simple passive might be ‘hit the same champion 3 times within X seconds to cause more damage’.

Champions are generally grouped into six categories:

• Marksmen buy items which improve their auto attack damage (a basic attack).
• Supports focus on items that improve their health and defences to be a frontline (someone who positions closest to the enemy team during fights in order to protect teammates and to create and engage fights), or items which help them to heal, shield, or defend others.
• Fighters look to buy items which help them do damage, but also to keep themselves healthy by healing themselves for a % of damage caused (for example). They tend to spend a lot of the game fighting solo against another solo opponent.
• Mages build items that improve their ability power, so their abilities do more damage.
• Assassins use items that help them kill squishy (vulnerable) champions quickly, known for producing quick burst damage.
• Tanks build items that give them massive improvements to health and defences.

They can then be grouped further by their effectiveness during the periods of the game:

• Early game: Strong during the early period, known as the ‘laning phase’, where there is fighting in small skirmishes, generally in equal numbers.
• Mid game: Period where teams start fighting for early objectives on the map; the start of grouping up as groups of 4 or 5.
• Late game: This is where most champions are now at or close to max level with their core items, and there are full-on 5v5 team fights.

The developer of League of Legends, Riot Games, works constantly to try and balance champions so all are viable. Taking into consideration the data from professional and general games, they adjust them to try and encourage an average 50% win rate for each. However, this doesn’t mean all champions are good against each other. For instance, Champion A might be strong against Champion B, but they get stomped by Champion C, and Champion C gets destroyed by Champion B. This comes into play later in the pick and ban phase, variable 3.

Additionally, champion popularity is also taken into account when considering balance changes. Win rates can be tied to champion usage and the skill of those playing; for instance, a champion might have a low win rate, but mainly due to being highly played by users who (quite frankly) are terrible.

Variables to note

• 171 champions
• 5 abilities (includes passive and ignores non-standard)
• 10 stat ranges per champion
• 6 champion categories

Variable 2: The map and player roles

The map the game is played on is square, and is split into 3 main lanes: Top, Mid and Bot. It is split diagonally (top left to bottom right) by the River – with the rest of the space in-between the lanes being the Jungle. The half of the square below and to the left of the River is the Blue team’s side, and the half above and to the right of the River is the Red team’s side.

In a normal game, side selection is random, whereas in the professional scene the team with the highest seed coming into the game generally gets to choose which side they are.

Figure 1. Summoner’s Rift, the map on which a standard game of League of Legends is played. Image adapted under Creative Commons License CC BY-SA 3.0.

The five players per team are distributed into the following positions:

• Top laner
• Jungler
• Mid laner
• Bot laner
• Support (who spends most of their time with the bot laner, but can roam to assist the other roles)

Champions tend to excel when played in a certain role, however you can technically play everyone everywhere, though you might get flamed (insulted, harassed, etc.) or called a griefer (someone purposely playing the game in a negative way).

In the professional game, each player is an expert in their role, and how it should be played, along with the champions that are suited to that lane. For the sake of simplicity, we’ll divide the total champion numbers by 5 to say that means each player is an expert in 34.2 champions. I refuse to round up or down.

While you might think, ‘Well that map sounds equal and balanced’, it turns out, you are super wrong, and you should feel bad. In the professional scene, Blue side generally has a win rate between 51–52%, although in last year’s tournament, Worlds 2024, the Blue win rate sky rocketed to 60% during the early stages! In the previous Worlds tournament, it was nearly 80%!

The map is mostly a flipped reflection, i.e. Team 1 top jungle matches Team 2 bottom jungle. While this seems fair, it does actually give blue slide a few advantages, such as easier access to the Baron Pit (the Baron is explained later) but generally it is also just a more natural way to play – going bottom left, to top right. Top right to bottom left makes for a more awkward process, especially when the game’s interface is mostly along the bottom edge covering a good portion of visibility.

Variables to note

• 2 sides
• 3 lanes + jungle
• 5 roles

Variable 3: The pick and ban phase

In the competitive scene, the order players choose a champion is very specific – and also involves banning champions out, which means both teams can’t play that champion. This is a good way to remove champions who perhaps are:

• too strong, so better if no-one plays it;
• a strong champion that someone on your team doesn’t play, so better to remove it; or
• is played by someone on the opposite team as an OTP (one trick pony), or the opponent is known for being able to affect the game strongly on that champion.

The order starts with the first ban phase:

• Blue Ban 1
  Red Ban 1
  Blue Ban 2
  Red Ban 2
  Blue Ban 3
  Red Ban 3

Then the first pick phase:

• Blue Pick 1
  Red Pick 1
  Red Pick 2
  Blue Pick 2
  Blue Pick 3
  Red Pick 3

Followed by the second ban phase:

• Red Ban 4
  Blue Ban 4
  Red Ban 5
  Blue Ban 5

And finally, the second pick phase:

• Red Pick 4
  Blue Pick 4
  Blue Pick 5
  Red Pick 5

Figure 2. The pick/ban order during champion selection. Image adapted under Creative Commons License CC BY-SA 3.0.

As you can see the pick orders are staggered; this is to try and ensure teams don’t become too unbalanced. Blue team get the opportunity to choose the strongest champion left in the pool; however Red team then gets to pick the next two. Depending on the meta (Most Effective Tactics Available), a team might prefer to be on blue or red side, i.e. if there is a singular strong champion who usually gets past the ban phase, you might want to ensure the chance to have them, thus pick Blue. Or if there is a range of strong picks, especially a pairing which works really well – you’d choose Red. As mentioned in Variable 2, Blue win rate is generally an advantage – one factor is due to a few champions who are very strong right now (meta picks), so Blue side tends to ban most of these out in the hope of being able to pick a remaining one.

The pick and ban structure also allows certain strategies.

Counter picking

In the pick phase, there is a tactic known as counter picking. Essentially, some champions are strong against certain other champions – or in turn weak against others. Therefore, you try to choose champions who are strong against the ones your opponents have chosen. The staggered order helps to ensure the team who picks second doesn’t get all the counter picks. The next strategy is also used to help with this …

Flex picking

As mentioned previously, while champions tend to excel in one role there are some which are strong in multiple roles, or at least capable enough to hold their own in them. Sometimes it becomes an advantage to choose one of these flex picks, rather than a strong meta champion in order to avoid being counter picked. You then get to decide when to reveal where this champion will go. It has been a strong tactic in the past to choose three flex picks in the first pick phase in order to really stump the opposing team. However, the negative of this tactic is that it requires multiple players to be competent at playing these champions.

Role banning

For example, if you are playing Blue side, and your Top Laner has already chosen their champion in the first pick phase, but the Red side’s Top Laner hasn’t chosen theirs, then you might choose to prioritise banning champions often played in Top during your second ban phase, to reduce the pool of viable Top Laners.

At the end of this, you’ll have two teams each with five champions. But it’s not just about choosing five strong champions and calling it a day: it’s about composition¹ as well, ensuring that your team of five don’t all have the same strength, but instead work together² to create a strong team. A team of five Mages, for example, wouldn’t work against a properly composed and balanced team which includes a Tank, Support and a Marksman. There is a bit of flexibility in how creative a team can be, but sticking to a well-rounded formula tends to lead to success.

So, is it game time yet? No, don’t be silly.

Variables to note

• LOTS! Let’s just throw a non-numerical value here … X

Variable 4: Runes, shards and summoner spells

In short, runes are extra effects you can assign to your champion; this is done during the Pick and Ban phase before the game starts.

Runes are split into 5 different categories, which are made from 4 different tier levels, the final tier being an ultimate rune.

• You have a primary category, where you can choose one from each tier – so 4 in total.
• You then have a secondary category, where you choose one from 3 tiers (no ultimate) – so 3 in total.

In total, there are 63 different runes – which all do different things!

Shards are another system, but much simpler. Three categories, each with 3 different options to choose from, which give you extra stats in areas such as health, cooldown reduction, adaptive power (which, depending on your champion, is either attack damage or ability power), etc.

Then there are summoner spells. These are extra skills players can take to help them perform. There are 9 different spells to choose from (although one is only for the Jungler as it helps them kill Jungle creeps) and each player can only have two.

Variables to note

• 5 rune categories with 63 runes in total, from which players can choose 7: 4 from their primary category and 3 from their secondary category
• 3 rune shard categories with 9 rune shards in total
• 9 summoner spells with a choice of 2 for each player

Lovely… so now the match can actually start. Yes!

And now we can go ‘no brain mode’ and enjoy the game right? Nein.

Variable 5: Items

In the game one of the main priorities is to earn gold, in order to buy items. This can be done in numerous ways, but the main ones are:

• Killing enemy champions
• Killing enemy minions, which are NPCs (non-playable characters) that march down the lanes
• Killing jungle minions (also NPCs)
• Taking turrets (fortifications which protect the lanes, each team has 3 per lane)

At the start of the game each player has 500 gold from which to buy a starter item. There are approximately 8 different starter items, each has its own stats and they are in certain cases special effects.

In the early game players buy item components; there are approximately 15 Basic Items which can make 48 Epic Items. These also have their own stats and in certain cases special effects.

And yep, that’s right – Basic and Epic item components can then be combined to create completed items classed as Legendary. There are about 103 of these, and by the end game each champion should have 5 of these. You can’t have the same item on a champion more than once, and some completed Items make it impossible to buy some others (if they are for a similar purpose, for instance).

All done? You know better than that.

Each champion also buys a pair of boots – which start as a basic pair, which are then upgraded to one of 7 variations.

Ok so now we…. Nope still not done, be quiet.

There’s a champion in the game called Ornn, who has a passive which allows him to upgrade a single item for each of his allies. There are 28 of those.

AND then there’s… why? Why are there all these items?

Well as you can imagine, each item has a different purpose. Some just purely boost damage, whether that’s attack damage, or ability power. Some improve your defences, things like health, armour or magic resist while some help you apply better shields or healing to your allies. Which ones you buy depends on the champion you play, AND which champions you are playing against. But also, players will have their own favourites of items they prefer to use – even if the statistics say not to, alas you can’t remove the human ego out of the equation!

Variables to note

• 6 item slots available for each champion
• 100+ Legendary items (plus Ornn variables if he’s selected as a champion)

Variable 6: Objectives

As mentioned previously – the aim of the game is to destroy your opponent’s Nexus. However, that is the final objective. To get there, you need to do objectives. Some are required; some are additional to help boost your chances of winning.

The main objectives are:

• Turrets: Each lane has 3 turrets. To get to the Nexus you need to at least destroy all 3 in a single lane. However, destroying more is beneficial for getting gold and improving your map state (i.e. having control of an area).
• Inhibitor: After destroying the 3rd turret in one lane, you have access to the opponent’s inhibitor for that lane. Destroying this allows your lane to spawn stronger minions.

Additional objectives which help your chances are:

• Dragons: Dragons spawn during the game; they can be four different types. The first two will be random types, but the following dragons will all be the same type and eventually become the ‘Soul’. The souls of the different types of dragon (Ocean, Cloud, Hextech and Infernal) offer different benefits. Whichever team gets four dragon kills (of any type combination) in total gets a dragon’s soul which offers a strong passive for the rest of the game. Oh, and depending on which dragon’s soul it is also changes the map in different ways.
• Grubs: Once per game, three small grub monsters spawn at the same time. Killing these will give a buff (enhancement) to your team which helps with the destruction of turrets.
• Herald: Herald is a large monster that spawns a while after the grubs, and removes the grubs if they’ve not been taken by this point. Should your team kill it, you can spawn it as an ally to help you easily destroy turrets.
• Baron: Baron spawns 25 minutes into the game. Should your team kill it, it gives each player alive at the time a temporary power boost, and the ability to power up your lane minions, to help you destroy turrets.
• Elder Dragon: After four dragons have been killed, the next dragon to spawn is Elder Dragon. Should your team kill this, it gives you a temporary power that will execute your opponents should you take them down to 20% health. This is a late game buff to help your team end the game.

The final objective:

• Nexus: Each team has a Nexus. Destroying your opponent’s Nexus ends the game in victory.

Variables to note

• Which dragon becomes soul, and which team gets it
• Which team gets grubs and how many
• Which team gets Herald
• Which team gets Barons or Elder Dragons

Variable 7: The game’s end(?)

It’s worth briefly mentioning, that while games in the main seasons usually comprise single games, in competitive tournaments (like the prementioned Worlds) there are also matches with best of three games, and best of five games. So, all of this happens over and over again! Additionally in subsequent games, the choice of side selection is decided by whichever team lost the previous game which means strategies change each time!

A recent introduction to the league is also ‘Fearless Draft’. In this system, during matches which span multiple games, whichever champions are played in Game 1, for instance, can’t be played in the later games. This creates a whole new dynamic and forces players to improve their pool of champions even further – along with making viewing more exciting and varied.

Variables to note

• You win or you lose
• Side selection for subsequent games

Disclaimer

There are further variables which could be discussed too, but the ones above are perhaps the most crucial to the standard gameplay.

The data

So here we are, at the crux of it all. Seven main variables, each with their own array of potential options, values and outcomes to take into consideration. It’s a swirling storm of statistics, strategy, psychology, and probability – like designing the world’s most complex game of rock, paper, scissors, but with 171 hands and a shifting rulebook.

Something that should be as simple as ‘win or lose’ is instead a story of data, and the integration of that data with player experience and knowledge to forge a path towards victory.

What fascinates me most is not just the numbers, but how those involved interpret them, bend them, and sometimes defy them, as with all sports – sometimes the better team, playing on the blue side and ahead for most the game, can lose. The data might suggest one path to victory, but the human element of intuition, creativity and stubbornness can rewrite the script entirely.

If you’re a mathematician, a gamer, or just someone curious about how numbers shape the world, I hope this blog has offered a glimpse into the beautiful chaos of esports. And maybe, just maybe, next time you hear someone mention League of Legends, you’ll see more than just a game – you’ll see a living, breathing equation in motion.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Ray Knight*

“Are we at the top?” might not sound like such an unusual question to hear from a child when you’re out and about, but as I cycled past a family and their dog in the park the other morning, I was intrigued by how the adult being questioned was going to answer. Anyway, as is so often the case when I hear snippets of conversation on the move, I thought that it would at best interrupt the flow of conversation, and at worst be considered rude, to stop and listen in, so I kept pedalling.

I’d like to pause briefly in my tale and invite you to think about the situations that could have prompted this question, given the location… (if you’re like me, you might make list, but otherwise just cast your mind around a park environment and consider the options). Maybe the pronoun ‘we’ ruled out trees or playground equipment from your musings, unless you see more families climbing things together than I do, but if not then you might have considered those alongside a hill, perhaps? In fact, they and I were on a bridge, and so the next questions I’d like to pose to you are how many different types of bridge have you come across and how many of those have what you would consider to be a ‘top’?

I thought about this before drawing some pictures (as a student of mine is fond of pointing out, “You love a sketch, Miss!”) and then decided to look up how many different types of bridge there are. Having found various internet sources that gave names to shapes I partially recognised, I couldn’t decide which of those types best described the bridge in question. I shared a photograph of ‘my’ bridge and pictures of the seven main structural types that I’d found on the internet, with a civil engineer I know, as I wasn’t sure which was the closest match or which particular elements were the most important in classifying it; for example, when identifying a triangle, most people first think of the property of having three sides, but to be certain a shape is a triangle, it’s also important that these sides are straight and that they form a closed shape. I wanted to know which properties give each bridge type its name and how to therefore classify one which didn’t exactly match any of the seven types I’d found described. Their answer was that ‘my’ bridge (shown below if you click on ‘Show / Hide photo’) is a double truss bridge, which I had assumed from the pictures I’d seen meant that the main span was flat. On further investigation, my assumption turned out to be wrong, so what my legs had told me, which was that more effort was increasingly required to ride across the bridge, up to a point, and then I could freewheel down the other side, must mean that there was a top, highest point or ‘apex’ (since we’re talking about the highest point of a geometric shape).

© Maciek Platek 2018.

I decided to ask AI for definitions of ‘top’, and this prompted me to realise other uses too. Below are some examples from our combined lists:

• The highest point or uppermost part of something (top of a mountain, building, or tree)
• The surface of an object (the top of the cube)
• Added to an existing word to modify the meaning (tabletop or countertop)
• The lid or cover of a container
• A position (“Put the lettuce on top of the cheese in the sandwich”)
• The highest position in a particular rank or chart (“The Top of the Pops”)
• A particular section of a data set (“The top 1% of…”)
• A piece of clothing worn on the upper part of the body
• A spinning toy that balances on a point at its base
• Peak performance state (being at the top of your game)
• Maximum volume or intensity (singing at the top of your lungs)

The versatility of the word is much wider than I’d first thought, and the context is often vital in understanding the concept being conveyed.

Back then to the question I overheard and the many answers it had made me consider: I went in search of photographs of the bridge to see if I could locate the ‘top’ via symmetry, both of the surface being walked/biked on, but also the visible highest point of the bridge (presumably at some unreachable point on the structure above us); I wondered if could locate the point at which I started to freewheel as the middle of the bridge, and therefore the highest I was going to get above the river on my bike. All these explorations generated joyful and interesting questions for me, arguably more so than if I’d overheard whatever response/answer was given in the moment. So as a colleague of mine often said, “Perhaps our role (as teachers) is to question answers, rather than answer questions”.

I leave you with “Do things with ‘tops’ always necessarily have ‘bottoms’?” and challenge you and your learners to think of as many examples and non-examples as you can!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Fran Watson*

What is the value of learning mathematics now that artificial intelligence (AI) can solve almost all the questions we throw at it? Will AI change the way we think of mathematics and the way we teach and learn it? Will learning mathematics even remain relevant in 10 years from now, in an age when AI will surely play a key role? These are all valid questions, regardless of whether one is a mathematician or an educator or not. Finding definite answers to such questions is a key challenge in times like these; governed by uncertainty, by the fact that many questions have no clear answers, and that most answers are questionable. It is challenging yet exciting at the same time though, just like mathematics! It is now that our human intelligence and character need to be fully activated to ensure we proceed confidently, and try to find answers that allow us to prosper as a civilisation. So, I cannot answer these big questions on behalf of everyone, but what I can and will do here is share my personal thoughts on mathematics in the age of AI, as a mathematician, an educator, and a human.

First, mathematics is here to stay, no matter how AI proliferates. Not necessarily as it is now, not necessarily as an independent subject, not necessarily in its current shape and form, but as something to be embraced and experienced, and as something to be learned as the Greek origins of the word imply. How will it be learned though? Not necessarily the same way as now, not necessarily in classrooms, not only through textbooks and assessments, and probably not just facilitated by a human. I am convinced that teaching and learning in general will change in theory and practice, sooner or later, as a result of AI and all the emerging technologies, all the new discoveries in various fields, and our own evolving understanding of the concepts of teaching and learning. A change in education models will surely accompany this, whether we are in favor of it or not. I am also certain that if we manage to perceive AI and technology as catalysts and not threats, and if we truly believe in our human intelligence and resilience and our ability to adapt while protecting our human characteristics, we can then conceive new resilient learning and education models, mostly human-led and AI- and technology-powered as a starting point, that take us to new levels of innovation, efficiency and progress.

Will mathematics remain relevant? Yes. Always. But again, what do we mean by mathematics and relevance here? Mathematics is not just a book or an assessment. It is more than a grade or a degree. Mathematics is relevant for how sublime it is, and we need to accentuate that. What do I mean by sublime? First, from a philosophical and aesthetic perspective, mathematics induces awe and exceeds the realm of senses; its infinite horizons and abstract beauty align with the scale and magnificence the word induces among philosophers, from Burke and the vastness beyond comprehension to Kant and triumph of reason. This somehow resonates with Kurt Gödel who said: “Either mathematics is too big for the human mind or the human mind is more than a machine.”1 For me personally, mathematics is SUBLIME in the way it embodies seven aspects: it is stimulating, ubiquitous, borderless, limitless, intriguing, monumental, and enduring. How, you ask?

Stimulating: Mathematics is not just memorising and applying formulas, sketching a graph, solving an equation, or finding the area of a 2D-shape, and we need to ensure it is not perceived as such. Nothing activates critical thinking, reasoning and problem-solving skills like mathematics. It pushes us to think about everything we deal with in real life from finances and measurements to more impalpable topics like the origin of the universe and the future of human intelligence, and even infinity and beyond. But not just that, mathematics teaches us to persevere while remaining patient, and to dream big while remaining humble. Maryam Mirzakhani was spot-on when she declared that “the beauty of mathematics only shows itself to more patient followers”.2 What is more humbling than the ‘simple’ prime numbers that still offer the most complex and challenging uncharted ramifications. Patience, and maybe AI, can help us explore these!

Ubiquitous: Mathematics is not just a ‘subject’ or a ‘course’ to be taught at school. While it has been tagged as such by many for a long time, for structural and practical reasons mainly, it might be time we rethink that. Purists may say that keeping it “independent” is an explicit acknowledgement of its importance, but is it? Maybe education systems should rethink how subjects are packaged and how content is delivered. Mathematics can (should?) be the element that beats the long-standing stagnation in the existing education models, which traditionally segregate subjects, with the help of new technologies, drawing upon how mathematics beats within so many other subjects that rely on mathematical concepts and reasoning to survive. Who said mathematics and computer science are totally independent, for example? Same for mathematics and physics or mathematics and medical sciences. Is teaching mathematics in context nowadays better or worse for learners? Joseph Fourier said: “Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them”.3 What if mathematics gets taught, at least partially, in that way; following a unifying approach?

Borderless: Mathematics is not to be confined in curricula and textbooks; it is not to be framed as a collection of topics and domains or a list of theorems and formulas. It is and should be free to roam, as Georg Cantor wanted it to be when he said “the essence of mathematics lies precisely in its freedom”.4 One of the beautiful things about technology, and mainly AI, when used carefully, is that it allows us to break human-made borders between education resources and brings almost everything to our fingertips: we can now access real-time data related to a medical condition for use in a statistics course; we can create a mathematical model for climate change based on relevant and recent information; we can visualise and touch 3D graphs and interact with all sorts of holograms; we can juggle equations and play with charts in various ways; and much more. When all of this is available, why stick to the book and the blackboard? Technology allows us to amplify how free-spirited mathematics can and should be. After all, mathematics is not just something we learn: it is something we live, and that should be at the core of how mathematics is experienced.

Limitless: Mathematics is not a finite set of components. It never ceases to amaze us day after day, and this will be the case for an infinite amount of time! Speaking of infinity, who can think of different ‘sizes’ of infinity but a mathematician? That being said, new technologies should be seen and used as tools in support of this journey of ours to ensure more brains are triggered, more mysteries are unraveled, more conjectures are proven, and even new domains and fields are created and explored, just like with DeepMind. After all, keep in mind that there are still many unsolved problems in mathematics that require deep thinking and much perseverance, so why not let AI help us tackle these quests while we move on to think of new mounts to climb? Andrew Wiles once said: “The definition of a good mathematical problem is the mathematics it generates rather than the problem itself.”5 Mathematics builds on itself, and the more we discover the more there will be left to uncover.

Intriguing: Mathematics is not boring and should not be so. It never ceases to push intelligence, human or artificial, to the limit, as it tackles a combination of structured and irregular numbers and objects, as it deals with systematic and chaotic systems and models, and as it includes logical and counterintuitive theorems and results. How fascinating to study a Koch snowflake, with its infinite length enclosing a finite area? How captivating to learn about Gabriel’s horn, with its infinite surface area and finite volume? This is the never-ending generosity of mathematics, always breaking the norms of reason and common sense, while being the guardian of logic! Mathematics is that magical mixture of certainty and conjectures: a special key that opens the most basic locks and the most challenging ones at once; it helps you add two digits and has the power to reveal the mysteries of the universe. But while this is so, we should not forget that there is still a stigma associated with mathematics for many, from numeracy to advanced level topics, and the fact that it is not presented and delivered as fun, attractive, and useful for all, adds to this stigma. We need to keep that gripping aspect of mathematics alive and to eliminate the stigma, and technology can assist with that.

Monumental: Mathematics is not just a supporting actor in a play led by other ‘fields’. Mathematics always had a tremendous role to play in humanity’s flourishing, and this has been the case to a massive extent from the beginning of times. Imagine a world without ‘0’, without ‘pi’, without ‘x’, without ‘i’, without ‘e’! All tiny on paper but huge in impact on human civilisation, and not only theoretically or in closed labs. Mathematics, whether explicitly or implicitly, even makes our world more beautiful to look at and live in; imagine architecture without the golden ratio, or nature without fractals and the Fibonacci sequence. This impact of mathematics will be further amplified with AI and new technologies; new discoveries, new proofs, new unexplored areas. But maybe it is time we rebrand mathematics to highlight its role and impact even more? If you ask a student nowadays if they prefer taking a course titled “Linear Algebra and Statistics” or one titled “Machine Learning and AI”, I believe I know what most of them would choose (someone to do a survey?), because mathematics is not marketed as a key component to all the transformations we are now witnessing in technology, nor to so many discoveries. Maybe the status of mathematics and its stardom is not debatable among scholars, but that is not necessarily the case in society or among students; this should change.

Enduring: Mathematics is not confined in an era or a timeline. It keeps on giving and history provides numerous examples of this. Mathematics evolves and expands, but it never stops being vital. Mathematics is always there, with its facts guiding how everything works whether we feel it or not, whether we know they are facts or not. As Erwin Schrödinger eloquently put it: “A mathematical truth is timeless, it does not come into being when we discover it”.6 Mathematics has always been a companion and a guide to discoveries and progress; from Thales, Euclid, and Hypatia; to Al Khawarizmi, Al-Battani and Al-Kashi; to Euler, Gauss, and Germain; to Ramanujan, Turing, and Mandelbrot; the list is long. Mathematics will remain alive as long as humanity exists, and beyond. It will remain relevant as long as we have problems to solve, patterns and relationships to understand, and things to measure and compare. Mathematics will remain a core element of this universe as long as curiosity is burning.

So, with or without AI, or maybe because of it, it is time to highlight how SUBLIME mathematics is! What do you say?

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Rachad Zaki*

Scarlett Howard

Humans have always been fascinated with space. We frequently question whether we are alone in the universe. If not, what does intelligent life look like? And how would aliens communicate?

The possibility of extraterrestrial life is grounded in scientific evidence. But the distances involved in travel between the stars are vast. If we do contact aliens, it would likely be via long distance communication, with our nearest neighbouring star being 4.4 light years away. Even being optimistic, it would likely take more than ten years for any round-trip communication.

How could that work when we have no shared language? Well, consider how we can engage with creatures here on Earth with minds quite alien to our own: bees.

Despite the vast differences in human and bee brains, both of us can do mathematics. As we argue in a new paper published in the journal Leonardo, our thought experiment lends weight to the idea that mathematics may form the basis for a “universal language,” which might one day be used to communicate between the stars.

Mathematics as the language of science

The idea of mathematics as universal is not new. Writing in the 17th century, Galileo Galilei described the universe as a grand book “written in the language of mathematics”.

Science fiction, too, has long explored the idea of mathematics as a universal language. In the 1985 novel and 1997 film Contact, extraterrestrials reach out to humans using a repeating sequence of prime numbers sent via radio signal.

In The Three-Body Problem, a novel by Liu Cixin adapted into a Netflix series, communication between aliens and humans to solve a mathematical problem occurs through a video game.

Mathematics also features in a 1998 novella by Ted Chiang called Story of Your Life, which was adapted into the 2016 film Arrival. It describes aliens with a non-linear experience of time and a correspondingly different formulation of mathematics.

Real scientific efforts at universal communication have also involved mathematics and numbers. The covers of the Golden Records, which accompanied the Voyager 1 and 2 space probes launched in 1977, are etched with mathematical and physical quantities to “communicate a story of our world to extraterrestrials”.

The 1974 Arecibo radio message beamed out into space consisted of 1,679 zeros and ones, ordered to communicate the numbers one to ten and the atomic numbers of the elements that make up DNA. In 2022, researchers developed a binary language designed to introduce extraterrestrials to human mathematics, chemistry, and biology.

This gold-aluminum cover was designed to protect the Voyager 1 and 2 ‘Sounds of Earth’ gold-plated records from micrometeorite bombardment, but also served a second purpose in providing the finder with a key to playing the record using binary arithmetic and numbers, as well as schematics to explain the process. NASA/JPL

How do we test a universal language without aliens?

A creature with two antennae, six legs, and five eyes may sound like an alien, but it also describes a bee. (Science fiction has of course imagined “insectoid” aliens.)

The ancestors of bees and humans diverged over 600 million years ago, yet we both possess communication, sociality, and some mathematical ability. Since parting ways, both honeybees and humans have independently developed effective, but different, means of communication and cooperation within complex societies.

Humans have developed language. Honeybees evolved the waggle dance – which communicates the location of food sources including distance, direction, angle from the Sun, and quality of the resource.

Due to our vast evolutionary separation from bees, as well as the differences between our brain sizes and structures, bees could be considered an insectoid alien model that exists right here on Earth. At least for the purposes of our thought experiment.

Bees and mathematics

In a series of experiments between 2016 and 2024, we explored the ability of bees to learn mathematics. We worked with freely flying honeybees that chose to regularly visit and participate in our outdoor maths tests to receive sugar water.

During the tests, bees showed evidence of solving simple addition and subtraction, categorising quantities as odd or even, and ordering quantities of items, including an understanding of “zero”. Bees even demonstrated the ability to link symbols with numbers, in a simple version of how humans learn Arabic and Roman numerals.

Bees have demonstrated the ability to learn simple arithmetic and can perform other numerical feats. Scarlett Howard

Despite the miniature brains of bees, they have demonstrated a rudimentary capacity to perform mathematics and learn to solve problems with quantities. Their mathematical ability involved learning to add and subtract one, which provides a launching pad to more abstract mathematics. The ability to add or subtract by one theoretically allows bees to represent all of the natural numbers.

If two species considered alien to each other – humans and honeybees – can perform mathematics, along with many other animals, then perhaps mathematics could form the basis of a universal language.

If there are extraterrestrial species, and they have sufficiently sophisticated brains, then our work suggests that they may have the capacity to do mathematics. A further question to be answered is whether different species will develop different approaches to mathematics, akin to dialects in language.

Such discoveries would also help to answer the question of whether mathematics is an entirely human construction, or if it is an a consequence of intelligence and thus, universal.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Scarlett Howard, Adrian Dyer & Andrew Greentree*

This AI-generated illustration is an example of how AI is at our fingertips. But mathematics lies at the heart of AI, and investment in these mathematical foundations will help Canada become a true global AI leader. (Adobe Stock), FAL

Artificial intelligence is everywhere. In fact, each reader of this article could have multiple AI apps operating on the very device displaying this piece. The image at the top of this article is also generated by AI.

Despite this, many mechanisms governing AI behaviour remain poorly understood, even to top AI experts. This leads to an AI race built upon costly scaling, both environmentally and financially, that is also dangerously unreliable.

Progress therefore depends not on escalating this race, but on understanding the principles underpinning AI. Mathematics lies at the heart of AI and investment in these mathematical foundations is the critical key to becoming a true global AI leader.

How AI shapes daily life

AI has rapidly become part of everyday life, not only in talking home devices and fun social media generation, but also in ways so seamless that many people don’t even notice its presence.

It provides the recommendations we see when browsing online and quietly optimizes everything from transit routes to home energy use.

Critical services rely on AI because it’s used in medical diagnosis, banking fraud detection, drug discovery, criminal sentencing, governmental services and health predictions, all areas where inaccurate outputs may have devastating consequences.

Problems, issues

Despite AI’s widespread use, serious and widely documented issues continue to showcase concerns around fairness, reliability and sustainability. Biases embedded in data and models can propagate discriminatory outcomes, from facial detection methods that perform well only on light skin tones to predictive tools that systematically disadvantage underrepresented groups.

These failures continue to be reported and range from racist outputs of ChatGPT and other chatbots to imaging tools that misidentify Barack Obama as white and biased criminal sentencing algorithms.

At the same time, the environmental and financial costs of deploying large-scale AI systems are growing at an extremely rapid pace.

If this trajectory continues, it will not only prove environmentally unsustainable, it will also concentrate access to these powerful AI tools to a few wealthy and influential entities with access to vast capital and massive infrastructure.

Teck Resources’ Highland Valley Copper Mine is seen near Logan Lake, B.C., in September 2025. Critical minerals like copper power everything from advanced semiconductors in chips to the massive data centres that train AI models. THE CANADIAN PRESS/Darryl Dyck

Why mathematics?

To address issues with a system, whether it’s fixing a car or ensuring reliability in an AI system, it’s crucial to understand how it works. A mechanic cannot fix or even diagnose why a car isn’t operating correctly without understanding how the engine works.

The “engine” for AI is mathematics. In the 1950s, scientists used ideas from logic and probability to teach computers how to make simple decisions. As technology advanced, so did the math, and tools from optimization, linear algebra, geometry, statistics and other mathematical disciplines became the backbone of what are now modern AI systems.

These methods are certainly modelled after aspects of the human brain, but despite the nomenclature of “neural networks” and “machine learning,” these systems are essentially giant math engines that carry out vast amounts of mathematical operations with parameters that were optimized using massive amounts of data.

This means improving AI is not just about continuously building bigger computers and using more data; it’s about deepening our understanding of the complex math that governs these systems. By recognizing how fundamentally mathematical AI really is, we can improve its fairness, reliability and sustainable scalability as it becomes an even larger part of everyday life.

Canada’s path forward

So what should Canada do next? Invest in the parts of AI that turn power into dependability. That means funding the science that makes AI systems predictable, auditable and efficient, so hospitals, banks, utilities and public agencies can adopt AI with confidence.

This is not a call for bigger servers; it’s a call for better science, where mathematics is the core scientific engine.

Artificial Intelligence Minister Evan Solomon waits to appear before the Standing Committee on Science and Research on Parliament Hill in Ottawa on Dec. 3, 2025. THE CANADIAN PRESS/Spencer Colby

Canada already has a national platform to advance this work: the mathematical sciences institutes the (Pacific Institute for the Mathematical Sciences, Fields Institute for Research in Mathematical Sciences, The Centre de recherches mathématiques, Atlantic Association for Research in the Mathematical Sciences, Banff International Research Station connect researchers across provinces and disciplines, convene collaborative programs and link academia with the public sector.

Together with Canada’s AI institutes (Mila, Vector, Amii) and CIFAR, this ecosystem strengthens both foundational and translational AI nationwide.

Canada’s standing in AI was built on decades of foundational research, work that preceded today’s large models and made them possible. Reinforcing that foundation would allow Canada to lead the next stage of AI development: models that are efficient rather than wasteful, transparent rather than opaque and trustworthy rather than brittle. Investing in mathematical research is not only scientifically essential, it is strategically wise and will strengthen national sovereignty.

The payoff is straightforward: AI that costs less to run, fails less often and earns more public trust. Canada can lead here, not by winning a computing power arms race, but by setting the scientific bar for how AI should work when lives, livelihoods and public resources are at stake.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Deanna Needell, Kristine Bauer & Ozgur Yilmaz*

Panther Media Global/Alamy

Mathematics is a “science which requires a great amount of imagination”, said the 19th-century Russian maths professor Sofya Kovalevskaya – a pioneering figure for women’s equality in this subject.

We all have an imagination, so I believe everyone has the ability to enjoy mathematics. It’s not just arithmetic but a magical mixture of logic, reasoning, pattern spotting and creative thinking.

Of course, more and more research also shows the benefits of doing puzzles like these for brain health and development. Canadian psychologist Donald Hebb’s theory of learning has come to be known as “when neurons fire together, they wire together” (which, by the way, is one of the guiding principles behind training large neural networks in AI). New pathways start to form which can build and maintain strong cognitive function.

What’s more, doing maths is often a collaborative endeavour – and can be a great source of fun and fulfilment when people work together on problems. Which brings me to these festive-themed puzzles, which can be tackled by the whole family. No formal training in maths is required, and no complicated formulas are needed to solve them.

I hope they bring you some moments of mindful relaxation this holiday season. You can read the answers (and my explanations for them) here.

Festive maths puzzlers

nestdesigns/Shutterstock

Puzzle 1: You are given nine gold coins that look identical. You are told that one of them is fake, and that this coin weighs less than the real ones. You are also given a set of old-fashioned balance scales that weigh groups of objects and show which group is heavier.

Question: What is the smallest number of weighings you need to carry out to determine which is the fake coin?

Puzzle 2: You’ve been transported back in time to help cook Christmas dinner. Your job is to bake the Christmas pie, but there aren’t even any clocks in the kitchen, let alone mobile phones. All you’ve got is two egg-timers: one that times exactly four minutes, and one that times exactly seven minutes. The scary chef tells you to put the pie in the oven for exactly ten minutes and no longer.

Question: How can you time ten minutes exactly, and avoid getting told off by the chef?

Dasha Efremova/Shutterstock

Puzzle 3: Having successfully cooked the Christmas pie, you are now entrusted with allocating the mulled wine – which is currently in two ten-litre barrels. The chef hands you one five-litre bottle and one four-litre bottle, both of which are empty. He orders you to fill the bottles with exactly three litres of wine each, without wasting a drop.

Question: How can you do this?

Puzzle 4: For the sake of this quiz, imagine there are not 12 but 100 days of Christmas. On the n-th day of Christmas, you receive £n as a gift, from £1 on the first day to £100 on the final day. In other words, far too many gifts for you to be able to count all the money!

Question: Can you calculate the total amount of money you have been given without laboriously adding all 100 numbers together?

(Note: a variation of this question was once posed to the German mathematician and astronomer Carl Friedrich Gauss in the 18th century.)

Puzzle 5: Here’s a Christmassy sequence of numbers. The first six in the sequence are: 9, 11, 10, 12, 9, 5 … (Note: the fifth number is 11 in some versions of this puzzle.)

Question: What is the next number in this sequence?

Garashchuk/Shutterstock

Puzzle 6: Take a look at the following list of statements:

Exactly one statement in this list of statements is false.

Exactly two statements in this list are false.

Exactly three statements in this list are false.

… and so on until:

Exactly 99 statements in this list are false.

Exactly 100 statements in this list are false.

Question: Which of these 100 statements is the only true one?

Puzzle 7: You are in a room with two other people, Arthur and Bob, who both have impeccable logic. Each of you is wearing a Christmas hat which is either red or green. Nobody can see their own hat but you can all see the other two.

You can also see that both Arthur’s and Bob’s hats are red. Now you are all told that at least one of the hats is red. Arthur says: “I do not know what colour my hat is.” Then Bob says: “I do not know what colour my hat is.”

Question: Can you deduce what colour your Christmas hat is?

Puzzle 8: There are three boxes under your Christmas tree. One contains two small presents, one contains two pieces of coal, and one contains a small present and a piece of coal. Each box has a label on it that shows what’s inside – but the labels have got mixed up, so every box currently has the wrong label on it. You are now told that you can open one box.

Question: Which box should you open, in order to then be able to switch the labels so that every label correctly shows the contents of its box?

Puzzle 9: Just before Christmas dinner, naughty Jack comes into the kitchen where there is one-litre bottle of orange juice and a one-litre bottle of apple juice. He decides to put a tablespoon of orange juice into the bottle of apple juice, then stirs it around so it’s evenly mixed.

But naughty Jill has seen what he did. Now she comes in, and takes a tablespoon of liquid from the bottle of apple juice and puts it into the bottle of orange juice.

Question: Is there now more orange juice in the bottle of apple juice, or more apple juice in the bottle of orange juice?

joto/Shutterstock

Puzzle 10: In Santa’s home town, all banknotes carry pictures of either Santa or Mrs Claus on one side, and pictures of either a present or a reindeer on the other. A young elf places four notes on a table showing the following pictures:

Santa   |   Mrs Claus   |   Present | Reindeer

Now an older, wiser elf tells him: “If Santa is on one side of the note, a present must be on the other.”

Question: Which notes must the young elf must turn over to confirm what the older elf says is true?

Bonus puzzle

If you need a festive tiebreaker, here’s a question that requires a little bit of algebra (and the formula “speed = distance/time”). It’s tempting to say this question can’t be solved because the distance is not known – but the magic of algebra should give you the answer.

Santa travels on his sleigh from Greenland to the North Pole at a speed of 30 miles per hour, and immediately returns from the North Pole to Greenland at a speed of 40 miles per hour

Tiebreaker: What is the average speed of Santa’s entire journey?

(Note: a non-Christmassy version of this question was posed by the American physicist Julius Sumner-Miller.)

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Neil Saunders*

Milad Fakurian / Unsplash

OpenAI chief executive Sam Altman – perhaps the most prominent face of the artificial intelligence (AI) boom that accelerated with the launch of ChatGPT in 2022 – loves scaling laws.

These widely admired rules of thumb linking the size of an AI model with its capabilities inform much of the headlong rush among the AI industry to buy up powerful computer chips, build unimaginably large data centres, and re-open shuttered nuclear plants.

As Altman argued in a blog post earlier this year, the thinking is that the “intelligence” of an AI model “roughly equals the log of the resources used to train and run it” – meaning you can steadily produce better performance by exponentially increasing the scale of data and computing power involved.

First observed in 2020 and further refined in 2022, the scaling laws for large language models (LLMs) come from drawing lines on charts of experimental data. For engineers, they give a simple formula that tells you how big to build the next model and what performance increase to expect.

Will the scaling laws keep on scaling as AI models get bigger and bigger? AI companies are betting hundreds of billions of dollars that they will – but history suggests it is not always so simple.

Scaling laws aren’t just for AI

Scaling laws can be wonderful. Modern aerodynamics is built on them, for example.

Using an elegant piece of mathematics called the Buckingham π theorem, engineers discovered how to compare small models in wind tunnels or test basins with full-scale planes and ships by making sure some key numbers matched up.

Those scaling ideas inform the design of almost everything that flies or floats, as well as industrial fans and pumps.

Another famous scaling idea underpinned the boom decades of the silicon chip revolution. Moore’s law – the idea that the number of the tiny switches called transistors on a microchip would double every two years or so – helped designers create the small, powerful computing technology we have today.

But there’s a catch: not all “scaling laws” are laws of nature. Some are purely mathematical and can hold indefinitely. Others are just lines fitted to data that work beautifully until you stray too far from the circumstances where they were measured or designed.

When scaling laws break down

History is littered with painful reminders of scaling laws that broke. A classic example is the collapse of the Tacoma Narrows Bridge in 1940.

The bridge was designed by scaling up what had worked for smaller bridges to something longer and slimmer. Engineers assumed the same scaling arguments would hold: if a certain ratio of stiffness to bridge length worked before, it should work again.

Instead, moderate winds set off an unexpected instability called aeroelastic flutter. The bridge deck tore itself apart, collapsing just four months after opening.

Likewise, even the “laws” of microchip manufacturing had an expiry date. For decades, Moore’s law (transistor counts doubling every couple of years) and Dennard scaling (a larger number of smaller transistors running faster while using the same amount of power) were astonishingly reliable guides for chip design and industry roadmaps.

As transistors became small enough to be measured in nanometres, however, those neat scaling rules began to collide with hard physical limits.

When transistor gates shrank to just a few atoms thick, they started leaking current and behaving unpredictably. The operating voltages could also no longer be reduced with being lost in background noise.

Eventually, shrinking was no longer the way forward. Chips have still grown more powerful, but now through new designs rather than just scaling down.

Laws of nature or rules of thumb?

The language-model scaling curves that Altman celebrates are real, and so far they’ve been extraordinarily useful.

They told researchers that models would keep getting better if you fed them enough data and computing power. They also showed earlier systems were not fundamentally limited – they just hadn’t had enough resources thrown at them.

But these are undoubtedly curves that have been fit to data. They are less like the derived mathematical scaling laws used in aerodynamics and more like the useful rules of thumb used in microchip design – and that means they likely won’t work forever.

The language model scaling rules don’t necessarily encode real-world problems such as limits to the availability of high-quality data for training, or the difficulty of getting AI to deal with novel tasks – let alone safety constraints or the economic difficulties of building data centres and power grids. There is no law of nature or theorem guaranteeing that “intelligence scales” forever.

Investing in the curves

So far, the scaling curves for AI look pretty smooth – but the financial curves are a different story.

Deutsche Bank recently warned of an AI “funding gap” based on Bain Capital estimates of a US$800 billion mismatch between projected AI revenues and the investment in chips, data centres and power that would be needed to keep current growth going.

JP Morgan, for their part, has estimated that the broader AI sector might need around US$650 billion in annual revenue just to earn a modest 10% return on the planned build-out of AI infrastructure.

We’re still finding out which kind of law governs frontier LLMs. The realities may keep playing along with the current scaling rules; or new bottlenecks – data, energy, users’ willingness to pay – may bend the curve.

Altman’s bet is that the LLM scaling laws will continue. If that’s so, it may be worth building enormous amounts of computing power because the gains are predictable. On the other hand, the banks’ growing unease is a reminder that some scaling stories can turn out to be Tacoma Narrows: beautiful curves in one context, hiding a nasty surprise in the next.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Nathan Garland*

 

Math teachers have to accommodate high school students’ different approaches to problem-solving. RJ Sangosti/MediaNews Group/The Denver Post via Getty Images

Among high school students and adults, girls and women are much more likely to use traditional, step-by-step algorithms to solve basic math problems – such as lining up numbers to add, starting with the ones place, and “carrying over” a number when needed. Boys and men are more likely to use alternative shortcuts, such as rounding both numbers, adding the rounded figures, and then adjusting to remove the rounding.

But those who use traditional methods on basic problems are less likely to solve more complex math problems correctly. These are the main findings of two studies our research team published in November 2025.

This new evidence may help explain an apparent contradiction in the existing research – girls do better at math in school, but boys do better on high-stakes math tests and are more likely to pursue math-intensive careers. Our research focuses not just on getting correct answers, but on the methods students use to arrive at them. We find that boys and girls approach math problems differently, in ways that persist into adulthood.

A possible paradox

In a 2016 study of U.S. elementary students, boys outnumbered girls 4 to 1 among the top 1% of scorers on a national math test. And over many decades, boys have been about twice as likely as girls to be among the top scorers on the SAT and AP math exams.

However, girls tend to be more diligent in elementary school and get better grades in math class throughout their schooling. And girls and boys across the grades tend to score similarly on state math tests, which tend to be more aligned with the school curriculum and have more familiar problems than the SAT or other national tests.

Beyond grades and test scores, the skills and confidence acquired in school carry far beyond, into the workforce. In lucrative STEM occupations, such as computer science and engineering, men outnumber women 3 to 1. Researchers have considered several explanations for this disparity, including differences in math confidence and occupational values, such as prioritizing helping others or making money. Our study suggests an additional factor to consider: gender differences in approaches to math problems.

When older adults think of math, they may recall memorizing times tables or doing the tedious, long-division algorithm. Memorization and rule-following can pay off on math tests focused on procedures taught in school. But rule-following has its limits and seems to provide more payoff among low-achieving than high-achieving students in classrooms.

More advanced math involves solving new, perplexing problems rather than following rules.

Math can be creative, not rote. AP Photo/Jacquelyn Martin

Differing strategies

In looking at earlier studies of young children, our research team was struck by findings that young boys use more inventive strategies on computation problems, whereas girls more often use standard algorithms or counting. We wondered whether these differences disappear after elementary school, or whether they persist and relate to gender disparities in more advanced math outcomes.

In an earlier study, we surveyed students from two high schools with different demographic characteristics to see whether they were what we called bold problem-solvers. We asked them to rate how much they agreed or disagreed with specific statements, such as “I like to think outside the box when I solve math problems.” Boys reported bolder problem-solving tendencies than girls did. Importantly, students who reported bolder problem-solving tendencies scored higher on a math problem-solving test we administered.

Our newer studies echo those earlier results but reveal more specifics about how boys and girls, and men and women, approach basic math problems.

Algorithms and teacher-pleasing

In the first study, we gave three questions to more than 200 high school students: “25 x 9 = ___,” “600 – 498 = ___,” and “19 + 47 + 31 = ___.” Each question could be solved with a traditional algorithm or with a mental shortcut, such as solving 25 x 9 by first multiplying 25 x 8 to get 200 and then adding the final 25 to get 225.

Regardless of their gender, students were equally likely to solve these basic computation items correctly. But there was a striking gender difference in how they arrived at that answer. Girls were almost three times as likely as boys – 52% versus 18% – to use a standard algorithm on all three items. Boys were far more likely than girls – 51% versus 15% – to never use an algorithm on the questions.

Girls were far more likely than boys to use an algorithm

When given three basic math problems, high school girls were three times more likely than boys to use a standard algorithm to solve all three. High school boys were nearly three times more likely than girls to use an alternative strategy for all three problems.

We suspected that girls’ tendency to use algorithms might stem from greater social pressure toward compliance, including complying with traditional teacher expectations.

So, we also asked all the students eight questions to probe how much they try to please their teachers. We also wanted to see whether algorithm use might relate to gender differences in more advanced problem-solving, so we gave students several complex math problems from national tests, including the SAT.

As we suspected, we found that girls were more likely to report a desire to please teachers, such as by completing work as directed. Those who said they did have that desire used the standard algorithm more often.

Also, the boys in our sample scored higher than the girls on the complex math problems. Importantly, even though students who used algorithms on the basic computation items were just as likely to compute these items correctly, algorithm users did worse on the more complex math problems.

Continuing into adulthood

In our second study, we gave 810 adults just one problem: “125 + 238 = ___.” We asked them to add mentally, which we expected would discourage them from using an algorithm. Again, there was no gender difference in answering correctly.

But 69% of women, compared to 46% of men, reported using the standard algorithm for their mental calculation, rather than using another strategy entirely.

We also gave the adults a more advanced problem-solving test, this time focused on probability-related reasoning, such as the chances that rolling a seven-sided die would result in an even number. Similar to our first study, women and those who used the standard algorithm on the computation problem performed worse on the reasoning test.

The importance of inventiveness

We identified some factors that may play a role in these gender differences, including spatial-thinking skills, which may help people develop alternate calculation approaches. Anxiety about taking tests and perfectionism, both more prevalent among women, may also be a factor.

We are also interested in the power of gender-specific social pressures on girls. National data has shown that young girls exhibit more studious behavior than do boys. And the high school girls we studied were more likely than boys to report they made a specific effort to meet teachers’ expectations.

More research definitely is needed to better understand this dynamic, but we hypothesize that the expectation some girls feel to be compliant and please others may drive teacher-pleasing tendencies that result in girls using algorithms more frequently than boys, who are more socialized to be risk-takers.

While compliant behavior and standard math methods often lead to correct answers and good grades in school, we believe schools should prepare all students – regardless of gender – for when they face unfamiliar problems that require inventive problem-solving skills, whether in daily life, on high-stakes tests or in math-intensive professions.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Sarah Lubienski, Colleen Ganley & Martha Makowski*

ThisIsEngineering/Pexels

As the artificial intelligence (AI) and quantum computing industries explode, trained STEM professionals are in high demand. Mathematics is foundational to these fields.

But mathematics is missing an important ingredient: people who are female or gender-diverse.

In New South Wales, for example, only one-third of high school graduates who complete mathematics at the highest level are female or gender-diverse. And when students choose university courses in December, a large proportion of these highly qualified people will step away from mathematics and STEM.

Australia cannot stay competitive by only accessing half of its young talent. By leaving mathematics early, young women and gender-diverse people limit their own career opportunities. Worse, the new technologies resulting from the current revolutions may not serve broader society well, if women and gender-diverse people are not involved in their development.

But at the University of Sydney over the past five years we have run a successful pilot program to reverse this trend – and to empower young women to make informed career choices. Better, the program is cheap to run and can be easily adopted elsewhere so mathematics – and the many industries it underpins – can be more diverse in ways that benefit everyone, regardless of their gender.

Declining enrolments

Before 2020, female and gender-diverse enrolments in advanced mathematics at the University of Sydney were in decline.

In 2020 the incoming cohort was nearly 80% male. Non-STEM directions offer attractive and important career options, and some movement between specialisations is expected. But a nosedive from 35% female students at the end of high school to 22% at the start of university indicates a problem.

Over five years, a team I lead piloted an intervention which has increased the ratio of female and gender-diverse students in advanced first-year mathematics from 22% to 30% – nearly back to the high school levels.

Our program consists of two components:

information, personalised invitations, and enrolment advice for incoming female and gender-diverse students, and a mentoring program for female and gender-diverse students who enrol in advanced mathematics.

Targeting the problem from year one

Before the start of semester, we compare first year enrolments with students’ high school certificates and majors. Like in high school, mathematics at the university is offered at multiple parallel levels.

When students are enrolled at a lower level than their background and major would justify, we send personalised emails encouraging them to switch to the advanced level. We hold a welcome event and multiple drop-in sessions, offering tailored advice.

In the mentoring program we match female and gender diverse advanced maths students with groups of eight to twelve peers of mixed year levels. Matching is based on timetables.

Each group is mentored by a senior (Honours or PhD) student, and an academic – at least one of whom is female or gender-diverse. Student mentors bring invaluable insight to the program, as they had walked in the mentees’ shoes only a few years before.

Each year 50–80 students participate in the program, roughly two-thirds of whom are first-year students.

Mentoring groups meet weekly for an hour: sometimes with both mentors, sometimes with the student mentor alone. Meeting topics are loosely structured around academic advice and sharing experiences.

Many groups develop their own agendas organically. The program does not focus on tutoring, though students enjoy discussing key mathematical techniques and concepts.

Fostering community and belonging

At the heart of the program is the opportunity to build community with peers, away from the pressure of assessments. While student feedback on the program is overall enthusiastic, it is a puzzle to maintain engagement with mentoring as semesters get hectic. It is difficult for students to prioritise community building when marks are on the line elsewhere.

We suspected the large drop in female and gender diverse enrolments at the transition to university is at least partly explained by these students’ lack of confidence in their mathematical abilities.

Research shows such insecurities disproportionately affect women. General messaging is ineffective in the face of self-doubt, so we aimed for a personalised but scalable approach.

The mentoring component fosters community and belonging. This combats isolation, provides ongoing support and enables long-term retention.

A low-cost solution

Our program is a low-cost solution that can be implemented in most academic contexts.

The first year of university is a place to start, but it is too late to fully address Australia’s pipeline problem. We can’t expect to have women and gender-diverse students participating in STEM at university in higher numbers than they did at the end of high school.

Similar programs could be put in place in high schools, and personal invitations can even be used to bring more girls to elementary school enrichment programs. This would help boost diverse and equitable participation in STEM from the roots.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Zsuzsanna Dancso*