Category: Skills and Learning

What is a Flipped Classroom?

Most teachers understand the “Chalk and Talk” or “Direct Instruction” method. The teacher begins by reviving what they did the day before, then continue with some new theories and concepts on the board, generally seeking student attention to work through the instances. Then once the maths students have the right set of notes from the board, they would use their textbook for a particular chapter, start solving the questions given by the teacher, and expectantly complete those tasks at home for homework.

As maths tutors, we are familiar that daily practice is significant. However, the students experience problems when practising, and their teacher isn’t there to assist them. The flipped classroom vision reorganizes what comes about at home and school compared to a more conventional plan. In short, the students will first find new content mainly independently, often as homework. Then in class, most of the time burnt out practising, finishing exercises, asking questions, and working on other activities in groups, with the teacher there to guide them.

Why do a Flipped Classroom?

Flipped classrooms permit one-on-one sessions with maths students who are practising, especially for the International Maths Olympiad, so we can move further in more effective directions. Change is challenging, so why do a flipped classroom? In short, change can be strenuous but productive. Bloom’s Two Sigma Problem demonstrates that a one-on-one session is the best method for teaching and learning.

How to Flip Maths Classroom?

Choose a topic to begin with, based on the timing, but you may select a topic that you believe matches the new strategy perfectly.

No matter your standard or plan for the organization, we suggest making a calendar to organize your unit before you begin.

It would be best if you had a simple outline of what lessons or concepts you will cover each day.

If you plan to create your own video sessions, you must figure out the best video recording practices.

Explain to students

If students are used to a specific teaching style and method, changing the pattern can also be an issue for them. It’s necessary to be clear with them about the switch that is taking place, why they’re happening, and what the students should anticipate in the outcome.

This is how one can flip for a maths classroom. Happy teaching!

It’s first period, Monday morning, and I’ve written a math problem on the board. But in front of me is a room full of blank stares and lowered heads.

I’ve got to get this class motivated, so I look to one of the students in the last row. “Hey Sally,” I ask, “did you watch the Giants game yesterday?”

“No, I’m a Jets fan. They’re way better.”

Another student, Sam, pipes up, “The Patriots are the best. They have Tom Brady.”

A few other kids chime in, throwing out their favorite teams. This goes on for a minute or two. Then I turn back to Sally and ask, “What was the score of the Jets game?”

“27–14. They beat the Dolphins.”

“Was it a close game?”

I get puzzled looks, but at least the whole class is looking at me now.

“No way! They won by 13, it was a blowout,” scoffs Sally.

Another student raises his hand, “Two more touchdowns and the Dolphins would have won. The quarterback threw an interception that should’ve been a touchdown.”

“Well, what did the Dolphins need to do in order to tie the game? A few field goals?”

Heads pop up. Now I’ve got their attention.

I start by writing on the board all the ways to score in football, and how many points a team gets for each: 6 for a touchdown, 3 for a field goal, 2 for a safety, and 1 (extra point kick) or 2 (scoring on a run or pass) for a conversion after a touchdown.

Excited, the students start discussing how the game could have been tied by the Dolphins. After a bit of back and forth, they agree that a touchdown, an extra point and two field goals is the best solution to tie the game. (6 + 1 + 3 + 3 = 13 points.) Though a field goal and five safeties would have been cool to see. (3 + 2 + 2 + 2 + 2 + 2 = 13 points.)

If you have a child who struggles with math, one thing you can do is connect math to his everyday life and interests. That real-world connection can get your child excited and engaged in learning.

Football is one of my favorite ways to motivate kids because there’s literally a new, fun math problem on every play. If you watch a game with your child, you can use this to your advantage.

Ask questions about score changes, yards gained or lost, time remaining, and so on. You’re not solving problems on a worksheet. This is a chance to be the coach or the announcer and analyze the game, all while reinforcing math concepts.

Want to try it out? Here are a few of my favorite conversation starters to get the football math flowing:

Situation #1: The score is Giants 17, Dolphins 21. There’s only enough time for the Giants to run one more play. Should the Giants go for a touchdown or kick a field goal?

By rapidly analysing large amounts of data and making accurate predictions, artificial intelligence (AI) tools could help to answer many long-standing research questions. For instance, they could help to identify new materials to fabricate electronics or the patterns in brain activity associated with specific human behaviours.

One area in which AI has so far been rarely applied is number theory, a branch of mathematics focusing on the study of integers and arithmetic functions. Most research questions in this field are solved by human mathematicians, often years or decades after their initial introduction.

Researchers at the Israel Institute of Technology (Technion) recently set out to explore the possibility of tackling long-standing problems in number theory using state-of-the-art computational models.

In a recent paper, published in the Proceedings of the National Academy of Sciences, they demonstrated that such a computational approach can support the work of mathematicians, helping them to make new exciting discoveries.

“Computer algorithms are increasingly dominant in scientific research, a practice now broadly called ‘AI for Science,'” Rotem Elimelech and Ido Kaminer, authors of the paper, told Phys.org.

“However, in fields like number theory, advances are often attributed to creativity or human intuition. In these fields, questions can remain unresolved for hundreds of years, and while finding an answer can be as simple as discovering the correct formula, there is no clear path for doing so.”

Elimelech, Kaminer and their colleagues have been exploring the possibility that computer algorithms could automate or augment mathematical intuition. This inspired them to establish the Ramanujan Machine research group, a new collaborative effort aimed at developing algorithms to accelerate mathematical research.

Their research group for this study also included Ofir David, Carlos de la Cruz Mengual, Rotem Kalisch, Wolfram Berndt, Michael Shalyt, Mark Silberstein, and Yaron Hadad.

“On a philosophical level, our work explores the interplay between algorithms and mathematicians,” Elimelech and Kaminer explained. “Our new paper indeed shows that algorithms can provide the necessary data to inspire creative insights, leading to discoveries of new formulas and new connections between mathematical constants.”

The first objective of the recent study by Elimelech, Kaminer and their colleagues was to make new discoveries about mathematical constants. While working toward this goal, they also set out to test and promote alternative approaches for conducting research in pure mathematics.

“The ‘conservative matrix field’ is a structure analogous to the conservative vector field that every math or physics student learns about in first year of undergrad,” Elimelech and Kaminer explained. “In a conservative vector field, such as the electric field created by a charged particle, we can calculate the change in potential using line integrals.

“Similarly, in conservative matrix fields, we define a potential over a discrete space and calculate it through matrix multiplications rather than using line integrals. Traveling between two points is equivalent to calculating the change in the potential and it involves a series of matrix multiplications.”

In contrast with the conservative vector field, the so-called conservative matrix field is a new discovery. An important advantage of this structure is that it can generalize the formulas of each mathematical constant, generating infinitely many new formulas of the same kind.

“The way by which the conservative matrix field creates a formula is by traveling between two points (or actually, traveling from one point all the way to infinity inside its discrete space),” Elimelech and Kaminer said. “Finding non-trivial matrix fields that are also conservative is challenging.”

As part of their study, Elimelech, Kaminer and their colleagues used large-scale distributed computing, which entails the use of multiple interconnected nodes working together to solve complex problems. This approach allowed them to discover new rational sequences that converge to fundamental constants (i.e., formulas for these constants).

“Each sequence represents a path hidden in the conservative matrix field,” Elimelech and Kaminer explained. “From the variety of such paths, we reverse-engineered the conservative matrix field. Our algorithms were distributed using BOINC, an infrastructure for volunteer computing. We are grateful to the contribution by hundreds of users worldwide who donated computation time over the past two and a half years, making this discovery possible.”

The recent work by the research team at the Technion demonstrates that mathematicians can benefit more broadly from the use of computational tools and algorithms to provide them with a “virtual lab.” Such labs provide an opportunity to try ideas experimentally in a computer, resembling the real experiments available in physics and in other fields of science. Specifically, algorithms can carry out mathematical experiments providing formulas that can be used to formulate new mathematical hypotheses.

“Such hypotheses, or conjectures, are what drives mathematical research forward,” Elimelech and Kaminer said. “The more examples supporting a hypothesis, the stronger it becomes, increasing the likelihood to be correct. Algorithms can also discover anomalies, pointing to phenomena that are the building-blocks for new hypotheses. Such discoveries would not be possible without large-scale mathematical experiments that use distributed computing.”

Another interesting aspect of this recent study is that it demonstrates the advantages of building communities to tackle problems. In fact, the researchers published their code online from their project’s early days and relied on contributions by a large network of volunteers.

“Our study shows that scientific research can be conducted without exclusive access to supercomputers, taking a substantial step toward the democratization of scientific research,” Elimelech and Kaminer said. “We regularly post unproven hypotheses generated by our algorithms, challenging other math enthusiasts to try proving these hypotheses, which when validated are posted on our project website. This happened on several occasions so far. One of the community contributors, Wolfgang Berndt, got so involved that he is now part of our core team and a co-author on the paper.”

The collaborative and open nature of this study allowed Elimelech, Kaminer and the rest of the team to establish new collaborations with other mathematicians worldwide. In addition, their work attracted the interest of some children and young people, showing them how algorithms and mathematics can be combined in fascinating ways.

In their next studies, the researchers plan to further develop the theory of conservative matrix fields. These matrix fields are a highly powerful tool for generating irrationality proofs for fundamental constants, which Elimelech, Kaminer and the team plan to continue experimenting with.

“Our current aim is to address questions regarding the irrationality of famous constants whose irrationality is unknown, sometimes remaining an open question for over a hundred years, like in the case of the Catalan constant,” Elimelech and Kaminer said.

“Another example is the Riemann zeta function, central in number theory, with its zeros at the heart of the Riemann hypothesis, which is perhaps the most important unsolved problem in pure mathematics. There are many open questions about the values of this function, including the irrationality of its values. Specifically, whether ζ(5) is irrational is an open question that attracts the efforts of great mathematicians.”

The ultimate goal of this team of researchers is to successfully use their experimental mathematics approach to prove the irrationality of one of these constants. In the future, they also hope to systematically apply their approach to a broader range of problems in mathematics and physics. Their physics-inspired hands-on research style arises from the interdisciplinary nature of the team, which combines people specialized in CS, EE, math, and physics.

“Our Ramanujan Machine group can help other researchers create search algorithms for their important problems and then use distributed computing to search over large spaces that cannot be attempted otherwise,” Elimelech and Kaminer added. “Each such algorithm, if successful, will help point to new phenomena and eventually new hypotheses in mathematics, helping to choose promising research directions. We are now considering pushing forward this strategy by setting up a virtual user facility for experimental mathematics,” inspired by the long history and impact of user facilities for experimental physics.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Ingrid Fadelli , Phys.org

Here is a list of some of the subsets of integral domains, along with the reasoning (a.k.a proofs) of why the bullseye below looks the way it does. Part 2 of this post will include back-pocket examples/non-examples of each.

Integral Domain: a commutative ring with 1 where the product of any two nonzero elements is always nonzero

Unique Factorization Domain (UFD): an integral domain where every nonzero element (which is not a unit) has a unique factorization into irreducibles

Principal Ideal Domain (PID): an integral domain where every ideal is generated by exactly one element

Euclidean Domain: an integral domain RR with a norm NN and a division algorithm (i.e. there is a norm NN so that for every a,b∈Ra,b∈R with b≠0b≠0, there are q,r∈Rq,r∈R so that a=bq+ra=bq+r with r=0r=0 or N(r)<N(b)N(r)<N(b))

Field: a commutative ring where every nonzero element has an inverse

Because… We can just choose the zero norm: N(r)=0N(r)=0 for all r∈Fr∈F.

Proof: Let FF be a field and define a norm NN so that N(r)=0N(r)=0 for all r∈Fr∈F. Then for any a,b∈Fa,b∈F with b≠0b≠0, we can writea=b(b−1a)+0.a=b(b−1a)+0.

Because… If I◃RI◃R is an arbitrary nonzero ideal in the Euclidean domain RR, then I=(d)I=(d), where d∈Id∈I such that dd has the smallest norm among all elements in II. Prove this using the division algorithm on dd and some a∈Ia∈I.

Proof: Let RR be a Euclidean domain with respect to the norm NN and let I◃RI◃R be an ideal. If I=(0)I=(0), then II is principle. Otherwise let d∈Id∈I be a nonzero element such that dd has the smallest norm among all elements in II. We claim I=(d)I=(d). That (d)⊂I(d)⊂I is clear so let a∈Ia∈I. Then by the division algorithm, there exist q,r∈Rq,r∈R so that a=dq+ra=dq+r with r=0r=0 or N(r)<N(d)N(r)<N(d). Then r=a−dq∈Ir=a−dq∈I since a,d∈Ia,d∈I. But my minimality of dd, this implies r=0r=0. Hence a=dq∈(d)a=dq∈(d) and so I⊂(d)I⊂(d).

Because…Every PID has the ascending chain condition (acc) on its ideals!* So to prove PID ⇒⇒ UFD, just recall that an integral domain RR is a UFD if and only if 1) it has the acc on principal ideals** and 2) every irreducible element is also prime.

Proof: Let RR be a PID. Then 1) RR has the ascending chain condition on principal ideals and 2) every irreducible element is also a prime element. Hence RR is a UFD.

Because… By definition.

Proof: By definition.

‍*Def: In general, an integral domain RR has the acc on its principal ideals if these two equivalent conditions are satisfied:

1. Every sequence I1⊂I2⊂⋯⊂⋯I1⊂I2⊂⋯⊂⋯ of principal ideals is stationary (i.e. there is an integer n0≥1n0≥1 such that In=In0In=In0 for all n≥n0n≥n0).
2. For every nonempty subset X⊂RX⊂R, there is an element m∈Xm∈X such that whenever a∈Xa∈X and (m)⊂(a)(m)⊂(a), then (m)=(a)(m)=(a).

**To see this, use part 1 of the definition above. If I1⊂I2⊂⋯I1⊂I2⊂⋯ is an acsending chain, consider their union I=⋃∞n=1InI=⋃n=1∞In. That guy must be a principal ideal (check!), say I=(m)I=(m). This implies that mm must live in some In0In0  for some n0≥1n0≥1 and so I=(m)⊂In0I=(m)⊂In0. But since II is the union, we have for all n≥n0n≥n0(m)=I⊃In⊃In0=(m).(m)=I⊃In⊃In0=(m).Voila!

Every field FF is a PID

because the only ideals in a field are (0)(0) and F=(1)F=(1)! And every field is vacuously a UFD since all elements are units. (Recall, RR is a UFD if every non-zero, non-invertible element (an element which is not a unit) has a unique factorzation into irreducibles).

‍In an integral domain, every maximal ideal is also a prime ideal. 

(Proof: Let RR be an integral domain and M◃RM◃R a maximal ideal. Then R/MR/M is a field and hence an integral domain, which implies M◃RM◃R is a prime ideal.)

Butut the converse is not true (see counterexample below). However, the converse is true in a PID because of the added structure!

(Proof: Let RR be a PID and (p)◃R(p)◃R a prime ideal for some p∈Rp∈R. Then pp is a prime – and hence an irreducible – element (prime ⇔⇔ irreducible in PIDs). Since in an integral domain a principal ideal is maximal whenever it is generated by an irreducible element, we conclude (p)(p) is maximal.)

This suggests that if you want to find a counterexample – an integral domain with a prime ideal which is not maximal – try to think of a ring which is not a PID:   In Z[x]Z[x], consider the ideal (p)(p) for a prime integer pp. Then (p)(p) is a prime ideal, yet it is not maximal since(p)⊂(p,x)⊂Z[x].(p)⊂(p,x)⊂Z[x].

‍If FF is a field, then F[x]F[x] – the ring of polynomials in xx with coefficients in FF – is a Euclidean domain with the norm N(p(x))=degp(x)N(p(x))=deg⁡p(x) where p(x)∈F[x]p(x)∈F[x].

By the integral domain hierarchy above, this implies every ideal in F[x]F[x] is of the form (p(x))(p(x)) (i.e. F[x]F[x] is a PID) and every polynomial can be factored uniquely into a product of prime polynomials (just like the integers)! The next bullet gives an “almost converse” statement.

‍If R[x]R[x] is a PID, the RR must be a field.

To see this, simply observe that R⊂R[x]R⊂R[x] and so RR must be an integral domain (since a subset of a integral domain inherets commutativity and the “no zero divisors” property). Since R[x]/(x)≅RR[x]/(x)≅R, it follows that R[x]/(x)R[x]/(x) is also an integral domain. This proves that (x)(x) is a prime ideal. But prime implies maximal in a PID! So R[x]/(x)R[x]/(x) – and therefore RR – is actually a field.

• This is how we know, for example, that Z[x]Z[x] is not a PID (in the counterexample a few bullets up) – ZZ is not a field!

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

*Credit for article given to Tai-Danae Bradley*

A maths problem previously tackled with the help of a computer, which produced a proof the size of Wikipedia, has now been cut down to size by a human. Although it is unlikely to have practical applications, the result highlights the differences between two modern approaches to mathematics: crowdsourcing and computers.

Terence Tao of the University of California, Los Angeles, has published a proof of the Erdős discrepancy problem, a puzzle about the properties of an infinite, random sequence of +1s and -1s. In the 1930s, Hungarian mathematician Paul Erdős wondered whether such a sequence would always contain patterns and structure within the randomness.

One way to measure this is by calculating a value known as the discrepancy. This involves adding up all the +1s and -1s within every possible sub-sequence. You might think the pluses and minuses would cancel out to make zero, but Erdős said that as your sub-sequences got longer, this sum would have to go up, revealing an unavoidable structure. In fact, he said the discrepancy would be infinite, meaning you would have to add forever, so mathematicians started by looking at smaller cases in the hopes of finding clues to attack the problem in a different way.

Last year, Alexei Lisitsa and Boris Konev of the University of Liverpool, UK used a computer to prove that the discrepancy will always be larger than two. The resulting proof was a 13 gigabyte file – around the size of the entire text of Wikipedia – that no human could ever hope to check.

Helping hands

Tao has used more traditional mathematics to prove that Erdős was right, and the discrepancy is infinite no matter the sequence you choose. He did it by combining recent results in number theory with some earlier, crowdsourced work.

In 2010, a group of mathematicians, including Tao, decided to work on the problem as the fifth Polymath project, an initiative that allows professionals and amateurs alike to contribute ideas through SaiBlogs and wikis as part of mathematical super-brain. They made some progress, but ultimately had to give up.

“We had figured out an interesting reduction of the Erdős discrepancy problem to a seemingly simpler problem involving a special type of sequence called a completely multiplicative function,” says Tao.

Then, in January this year, a new development in the study of these functions made Tao look again at the Erdős discrepancy problem, after a commenter on his SaiBlog pointed out a possible link to the Polymath project and another problem called the Elliot conjecture.

Not just conjecture

“At first I thought the similarity was only superficial, but after thinking about it more carefully, and revisiting some of the previous partial results from Polymath5, I realised there was a link: if one could prove the Elliott conjecture completely, then one could also resolve the Erdős discrepancy problem,” says Tao.

“I have always felt that that project, despite not solving the problem, was a distinct success,” writes University of Cambridge mathematician Tim Gowers, who started the Polymath project and hopes that others will be encouraged to participate in future. “We now know that Polymath5 has accelerated the solution of a famous open problem.”

Lisitsa praises Tao for doing what his algorithm couldn’t. “It is a typical example of high-class human mathematics,” he says. But mathematicians are increasingly turning to machines for help, a trend that seems likely to continue. “Computers are not needed for this problem to be solved, but I believe they may be useful in other problems,” Lisitsa says.

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

*Credit for article given to Jacob Aron*

 

When choosing the perfectly sized table to do your jigsaw puzzle on, work out the area of the completed puzzle and multiply it by 1.73.

People may require a larger table if they like to lay all the pieces out at the start, rather than keeping them in the box or in piles

How large does your table need to be when doing a jigsaw puzzle? The answer is the area of the puzzle when assembled multiplied by 1.73. This creates just enough space for all the pieces to be laid flat without any overlap.

“My husband and I were doing a jigsaw puzzle one day and I just wondered if you could estimate the area that the pieces take up before you put the puzzle together,” says Madeleine Bonsma-Fisher at the University of Toronto in Canada.

To uncover this, Bonsma-Fisher and her husband Kent Bonsma-Fisher, at the National Research Council Canada, turned to mathematics.

Puzzle pieces take on a range of “funky shapes” that are often a bit rectangular or square, says Madeleine Bonsma-Fisher. To get around the variation in shapes, the pair worked on the basis that all the pieces took up the surface area of a square. They then imagined each square sitting inside a circle that touches its corners.

By considering the area around each puzzle piece as a circle, a shape that can be packed in multiple ways, they found that a hexagonal lattice, similar to honeycomb, would mean the pieces could interlock with no overlap. Within each hexagon is one full circle and parts of six circles.

They then found that the area taken up by the unassembled puzzle pieces arranged in the hexagonal pattern would always be the total area of the completed puzzle – calculated by multiplying its length by its width – multiplied by the root of 3, or 1.73.

This also applies to jigsaw puzzle pieces with rectangular shapes, seeing as these would similarly fit within a circle.

While doing a puzzle, some people keep pieces that haven’t yet been used in the box, while others arrange them in piles or lay them on a surface, the latter being Madeleine Bonsma-Fisher’s preferred method. “If you really want to lay all your pieces out flat and be comfortable, your table should be a little over twice as big as your sample puzzle,” she says.

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

*Credit for article given to Chen Ly*

While adding up your grocery bill in the supermarket, you’re probably not thinking how important or sophisticated our number system is.

But the discovery of the present system, by unknown mathematicians in India roughly 2,000 years ago – and shared with Europe from the 13th century onwards – was pivotal to the development of our modern world.

Now, what if our “decimal” arithmetic, often called the Indo-Arabic system, had been discovered earlier? Or what if it had been shared with the Western world earlier than the 13th century?

First, let’s define “decimal” arithmetic: we’re talking about the combination of zero, the digits one through nine, positional notation, and efficient rules for arithmetic.

“Positional notation” means that the value represented by a digit depends both on its value and position in a string of digits.

Thus 7,654 means:

(7 × 1000) + (6 × 100) + (5 × 10) + 4 = 7,654

The benefit of this positional notation system is that we need no new symbols or calculation schemes for tens, hundreds or thousands, as was needed when manipulating Roman numerals.

While numerals for the counting numbers one, two and three were seen in all ancient civilisations – and some form of zero appeared in two or three of those civilisations (including India) – the crucial combination of zero and positional notation arose only in India and Central America.

Importantly, only the Indian system was suitable for efficient calculation.

Positional arithmetic can be in base-ten (or decimal) for humans, or in base-two (binary) for computers.

In binary, 10101 means:

(1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + 1

Which, in the more-familiar decimal notation, is 21.

The rules we learned in primary school for addition, subtraction, multiplication and division can be easily extended to binary.

The binary system has been implemented in electronic circuits on computers, mostly because the multiplication table for binary arithmetic is much simpler than the decimal system.

Of course, computers can readily convert binary results to decimal notation for us humans.

As easy as counting from one to ten

Perhaps because we learn decimal arithmetic so early, we consider it “trivial”.

Indeed the discovery of decimal arithmetic is given disappointingly brief mention in most western histories of mathematics.

In reality, decimal arithmetic is anything but “trivial” since it eluded the best minds of the ancient world including Greek mathematical super-genius Archimedes of Syracuse.

Archimedes – who lived in the 3rd century BCE – saw far beyond the mathematics of his time, even anticipating numerous key ideas of modern calculus. He also used mathematics in engineering applications.

Nonetheless, he used a cumbersome Greek numeral system that hobbled his calculations.

Imagine trying to multiply the Roman numerals XXXI (31) and XIV (14).

First, one must rewrite the second argument as XIIII, then multiply the second by each letter of the first to obtain CXXXX CXXXX CXXXX XIIII.

These numerals can then be sorted by magnitude to arrive at CCCXXXXXXXXXXXXXIIII.

This can then be rewritten to yield CDXXXIV (434).

(For a bit of fun, try adding MCMLXXXIV and MMXI. First person to comment with the correct answer and their method gets a jelly bean.)

Thus, while possible, calculation with Roman numerals is significantly more time-consuming and error prone than our decimal system (although it is harder to alter the amount payable on a Roman cheque).

History lesson

Although decimal arithmetic was known in the Arab world by the 9th century, it took many centuries to make its way to Europe.

Italian mathematician Leonardo Fibonacci travelled the Mediterranean world in the 13th century, learning from the best Arab mathematicians of the time. Even then, it was several more centuries until decimal arithmetic was fully established in Europe.

Johannes Kepler and Isaac Newton – both giants in the world of physics – relied heavily on extensive decimal calculations (by hand) to devise their theories of planetary motion.

In a similar way, present-day scientists rely on massive computer calculations to test hypotheses and design products. Even our mobile phones do surprisingly sophisticated calculations to process voice and video.

But let us indulge in some alternate history of mathematics. What if decimal arithmetic had been discovered in India even earlier, say 300 BCE? (There are indications it was known by this date, just not well documented.)

And what if a cultural connection along the silk-road had been made between Indian mathematicians and Greek mathematicians at the time?

Such an exchange would have greatly enhanced both worlds, resulting in advances beyond the reach of each system on its own.

For example, a fusion of Indian arithmetic and Greek geometry might well have led to full-fledged trigonometry and calculus, thus enabling ancient astronomers to deduce the laws of motion and gravitation nearly two millennia before Newton.

In fact, the combination of mathematics, efficient arithmetic and physics might have accelerated the development of modern technology by more than two millennia.

It is clear from history that without mathematics, real progress in science and technology is not possible (try building a mobile phone without mathematics). But it’s also clear that mathematics alone is not sufficient.

The prodigious computational skills of ancient Indian mathematicians never flowered into advanced technology, nor did the great mathematical achievements of the Greeks, or many developments in China.

On the other hand, the Romans, who were not known for their mathematics, still managed to develop some impressive technology.

But a combination of advanced mathematics, computation, and technology makes a huge difference.

Our bodies and our brains today are virtually indistinguishable from those of ancient times.

With the earlier adoption of Indo-Arabic decimal arithmetic, the modern technological world of today might – for better or worse – have been achieved centuries ago.

And that’s something worth thinking about next time you’re out grocery shopping.

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

*Credit for article given to Jonathan Borwein (Jon)*

The third diagonal column in Pascal’s Triangle (r = 2 in the usual way of labeling and numbering) consists of the triangular numbers (1, 3, 6, 10, …) – numbers that can be arranged in 2-dimensional triangular patterns. The fourth column of Pascal’s triangle gives us triangular-based pyramidal numbers (1, 4, 10, 20, …), built by stacking the triangular numbers. The columns further out give “higher dimensional” triangular numbers that arise from stacking the triangular numbers from the previous dimension.

It is not by coincidence that the triangular and higher-dimensional triangular numbers appear in Pascal’s Triangle. If you think about layering of polygonal numbers in terms of equations, you get

In the above equation p^d_(k,n) is the nth k-polygonal number of dimension d. Triangular numbers are the 3-polygonal numbers of dimension 2, square numbers are the 4-polygonal numbers of dimension 2, “square based pyramidal numbers” would be denoted as p^3_(4,n).
from the sum above, you can obtain this equation:

Which looks very much like the Pascal Identity C(n,r) = C(n-1,r-1) + C(n-1,r), except for some translation of the variables. To be precise, if we consider the case where k=3 and use r = d and n‘ = n+d-1 we can translate the triangular numbers into the appropriate positions in Pascal’s Triangle.

Along with the definitions for the end columns, the Pascal Identity allows us to generate the whole triangle. This suggests the following strategy for calculating the higher k-Polygonal numbers: create a modified Pascal’s Triangle whose first column is equal to k-2 (instead of 1), and whose last column is equal to 1 (as usual). This modified Pascal’s Triangle is generated using these initial values and the usual Pascal Identity.

Here is an example with k=5, which sets the first column values equal to 3 (except for the top value, which we keep as 1) and yields the pentagonal numbers (column 3) and the higher pentagonal numbers.

The formula for these modified Pascal Triangles is given by this equation:

If we apply the change of variables mentioned above, we can obtain this general formula for the higher polygonal numbers in terms of combinations:

This formula illustrates how polygonal numbers are built out of triangular numbers. It says that the nth d-dimensional k-polygonal number is equal to the nth d-dimensional triangular number, plus (k-3) copies of the n-1 d-dimensional triangular number. This is a little easier to understand when you forget about the higher-dimensions and look at the regular 2-dimensional polygonal number.

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

*Credit for article given to dan.mackinnon*

 

Today I’d like to share an idea. It’s a very simple idea. It’s not fancy and it’s certainly not new. In fact, I’m sure many of you have thought about it already. But if you haven’t—and even if you have!—I hope you’ll take a few minutes to enjoy it with me. Here’s the idea:

So simple! But we can get a lot of mileage out of it.

To start, I’ll be a little more precise: every matrix corresponds to a weighted bipartite graph. By “graph” I mean a collection of vertices (dots) and edges; by “bipartite” I mean that the dots come in two different types/colors; by “weighted” I mean each edge is labeled with a number.

The graph above corresponds to a 3×23×2 matrix MM. You’ll notice I’ve drawn three greengreen dots—one for each row of MM—and two pinkpink dots—one for each column of MM. I’ve also drawn an edge between a green dot and a pink dot if the corresponding entry in MM is non-zero.

For example, there’s an edge between the second green dot and the first pink dot because M21=4M21=4, the entry in the second row, first column of MM, is not zero. Moreover, I’ve labeled that edge by that non-zero number. On the other hand, there is no edge between the first green dot and the second pink dot because M12M12, the entry in the first row, second column of the matrix, is zero.

Allow me to describe the general set-up a little more explicitly.

Any matrix MM is an array of n×mn×m numbers. That’s old news, of course. But such an array can also be viewed as a function M:X×Y→RM:X×Y→R where X={x1,…,xn}X={x1,…,xn} is a set of nn elements and Y={y1,…,ym}Y={y1,…,ym} is a set of mm elements. Indeed, if I want to describe the matrix MM to you, then I need to tell you what each of its ijijth entries are. In other words, for each pair of indices (i,j)(i,j), I need to give you a real number MijMij. But that’s precisely what a function does! A function M:X×Y→RM:X×Y→R associates for every pair (xi,yj)(xi,yj) (if you like, just drop the letters and think of this as (i,j)(i,j)) a real number M(xi,yj)M(xi,yj). So simply write MijMij for M(xi,yj)M(xi,yj).

Et voila. A matrix is a function.

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

*Credit for article given to Tai-Danae Bradley*

Brian Hayes recently provided some evidence that there are still many out there who are confounded by the Monty Hall problem.

The Monty Hall problem, perhaps the best-known counter-intuitive probability problem, gets a nice treatment in Jeffery Rosenthal’s Struck by Lightning: The Curious World of Probabilities, and is also explained well (perhaps better) in Mark Haddon’s novel The Curious Incident of the Dog in the Night-time. Professor Rosenthal has some further Monty Hall explanations here.

I just found an alternate version of the problem in Martin Gardner’s The Second Scientific American Book of Mathematical Puzzles and Diversions (published also by Penguin in a slightly different form as More Mathematical Puzzles and Diversions). In the chapter “Probability and Ambiguity” (chapter 19 in both versions of the book), Gardner describes the problem of the three prisoners. Here is a condensed description of the problem:

Three prisoners, A, B, and C are in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. Finding out that one is to be released, prisoner A begs the warden to let him know the identity of one of the others who is going to be executed. “If B is to be pardoned, give me C’s name. If C is to be pardoned, give me B’s name. And if I’m to be pardoned, flip a coin to decide whether to name B or C.”

The warden tells A that B is to be executed. Prisoner A is pleased because he believes that his probability of surviving has gone up from 1/3 to 1/2. Prisoner A secretly tells C the news, who is also happy to hear it, believing that his chance of survival has also risen to 1/2.

Are A and C correct? No. Prisoner A’s probability of surviving is still 1/3, but prisoner C’s probability of receiving the pardon is 2/3.

It is reasonably easy to see that the 3 prisoners problem is the same as the Monty Hall problem. Seeing the problem in this different formulation might help those who continue to struggle with it.

It is a nice activity to simulate both the 3 prisoners problem and the Monty Hall problem in Fathom – try it and confirm the surprising results that the “second prisoner” is pardoned 2/3 of the time, and that 2/3 of the time, the winning curtain is not the one you selected first.

There are many, many, ways to write these simulations. Here are the attributes and formulas for a Fathom implementation of the three prisoners:

The table below (click on it to see a larger version) shows a separate simulation for the Monty Hall problem. Here we are assuming three curtains “1”, “2”, and “3”, one of which has a prize behind it. You pick one, and then Monty reveals the contents behind one of the other curtains (the curtain with the prize behind it is not shown). In the game, you have the option of switching your choice for the curtain that has not been revealed.

After creating the attributes, you can “run the simulation” by adding data to the collection (Collection->New Cases…), the more the better.

Incidentally, Gardner’s use of A, B, and C reminds me of Stephen Leacock’s “A, B, and C: The Human Element in Mathematics.”

Addendum

A quick search shows that the connection between the Monty Hall problem and the Three Prisoners is well known (see the wikipedia entries on Monty Hall and the Three Prisoners), and that both are alternate formulations of an older problem, known as Bertrand’s box paradox.

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

*Credit for article given to dan.mackinnon*