Category: Uncategorized

A simple experiment and mathematical model suggest that when you snack on pistachios, you may need a surprisingly large bowl to accommodate the discarded shells.

Shelling your favourite snack nuts just got a lot easier: physicists have worked out the exact size of bowl to best fit discarded pistachio shells.

Ruben Zakine and Michael Benzaquen at École Polytechnique in Paris often find themselves discussing science in the cafeteria while eating pistachios. Naturally, they began wondering about the mathematics behind storing their snack refuse.

The researchers stuffed 613 pistachios into a cylindrical container to determine “packing density”, or the fraction of space taken up by whole nuts in their shells. Separately, they measured the packing density of the shells alone. In one experiment setup, the researchers poured the shells into a container and let them fall as they may, and in another they shook them into a denser, more efficient configuration.

Without shaking, the shells had about 73 per cent of the original packing density. Shaking decreased this number to 57 per cent. This suggests that, with any pistachio container, an additional half-sized container will hold shell refuse as long you occasionally shake the container while eating.

Zakine and Benzaquen backed up their findings by modelling pistachios as ellipsoids – three-dimensional shapes resembling squashed spheres – and their shells as hollow half-spheres and calculated their packing densities based on mathematical rules. These results confirmed the real-life experiments and suggested that the same ratios would work for other container shapes.

Despite these similarities, the researchers found about a 10 per cent discrepancy between the calculations and the real-life measurements. Zakine says that this is not surprising because pistachios are not perfect ellipsoids and have natural variations in shape. More broadly, it is tricky to calculate how best to pack objects into containers. So far, mathematics researchers have only had luck with doing calculations for spheres, like marbles, and uniform shapes like M&M’s, he says.

Going forward, the researchers want to run more complex calculations on a computer. But for now, they are looking forward to fielding mathematical questions whenever they serve pistachios at dinner parties.

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

*Credit for article given to Karmela Padavic-Callaghan*

Catering for students’ learning needs is something we all aim to do. But it can be challenging. Is it just about differentiation? What is the best way to differentiate? How do we put it into practice? Let’s explore some ideas, strategies and tips.

Differentiation

When you hear the word differentiation, what do you think of? Ability groupings? Open-ended tasks? Educational consultant Jennifer Bowden from the Mathematical Association of Victoria believes differentiation involves teachers considering “a whole range of different pedagogies … and making choices about pedagogical approaches based on the students that they teach”. In a nutshell it comes down to knowing your students and how they learn, so you can cater for their needs.

Find out what students know

Assessment is key to discovering what your students know – and don’t know! You can assess students to find out what knowledge they have, the concepts they understand and the skills they can apply to tasks.

Data from this assessment can then be used as a starting point to plan what you will teach.

Find out how students learn

You can go further than just understanding what your students know. Delve deeper and think about; what are your students’ learning behaviours? What are their attitudes towards learning maths? How do they learn best?

It’s important to note that this Is not about learning styles. It’s about knowing how a student:

• thinks and feels about maths
• becomes engaged in a topic, or problem
• responds to certain scaffolds
• makes connections between concepts
• applies what they have learnt.

When you understand your students on this level you have a greater insight into knowing how to best build their knowledge and skills.

Putting it into practice

Once you know your students well you are better prepared to meet their learning needs, but there are still many aspects to think about. Let’s unpack this further.

Planning for instruction

Maths expert Jennifer Bowden promotes the use of the instructional model known as launch, explore, summarise.

• Launch– begin with a question or a task for students to complete or explore.
• Explore– during this stage the teacher supports students at their different levels. Students can work on the same task, but it can be differentiated to extend or give extra support where needed by scaffolding. You can plan for the learning to be done independently, or in small groups.
• Summarise– upon competition of the lesson or task the students come together to share what they have learnt.

In an excellent podcast on the Maths Hub, Jennifer explains this model in greater detail.

Open-ended tasks

These rich tasks provide differentiation by output. Essentially all students are working on the same, or similar task, and students reach various outcomes, according to their individual knowledge and skill application.

Grouping students

There are times when you can best meet students’ needs through grouping them in certain ways. When doing so, consider the purpose of the groupings, and ensure the groups are flexible.

• You should be clear about the specific purpose of your groupings. What needs are you addressing by grouping students together? Are you extending them? Providing consolidation? Are you supporting them to ‘catch up’ on learning they have missed? Or providing intervention?
• Student groupings should beflexible and change according to their purpose. Sometimes groups are ability based, so students can complete different tasks, at different levels. Sometimes groups have mixed abilities so that students can use their various skills and levels of knowledge to problem solve and use their reasoning skills.

Student agency

Giving students a voice by encouraging them to discuss their learning can help you to understand their individual needs. Ask students about their learning; what they know and want to know, if they are feeling challenged and what helps them to learn. This feedback can help you plan and deliver lessons that cater for all student needs.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Mathematics Hub

The 14-sided polygon Tile(1,1), on the left, is known as a weakly chiral aperiodic monotile — in other words, if tilings that mix unreflected and reflected tiles are forbidden, then it tiles only aperiodically. However, by modifying its edges, as shown in the centre and right, strictly chiral aperiodic monotiles called “spectres” are created that admit only non-periodic tilings. Credit: University of Waterloo

Recently, an international team of four, that includes Cheriton School of Computer Science professor Dr. Craig Kaplan, discovered a single shape that tiles the plane—an infinite, two-dimensional surface—in a pattern that can never be made to repeat.

The discovery mesmerized mathematicians, tiling enthusiasts and the public alike.

The shape, a 13-sided polygon they called “the hat,” is known to mathematicians as an aperiodic monotile or an “einstein,” the German words that mean “one stone.”

But the team’s most recent discovery has raised the bar once again. They found another shape, related to the first, that meets an even stricter definition. Dubbed the “specter,” the new shape tiles a plane in a pattern that never repeats without the use of mirror images of the shape. For this reason, it has also been called a “vampire einstein”—a shape that tiles aperiodically without requiring its reflection.

“Our first paper solved the einstein problem, but as the shape required reflection to tile aperiodically people raised a legitimate question: Is there a shape that can do what the hat does but without reflection,” Kaplan explains. “It was our good fortune that we found a shape that not only solves this subproblem, but also solved it so soon after the first paper.”

To mathematicians, the hat and its mirror image are a single shape, but in the physical world left-handed and right-handed shapes can behave differently. You can’t, for example, wear a right-handed glove on your left hand.

“If you tiled a large bathroom floor aperiodically with hat-shaped tiles that had been glazed on one side you would need hats and mirror images of hats,” Kaplan says.

But it was not this quibble that motivated the recent discovery.

The discovery of the vampire einstein began with the musings of David Smith, a retired print technician and self-described shape hobbyist from Yorkshire, England, whose curiosity months earlier led to the original einstein discovery.

“Dave emailed us a couple of days after our hat paper went online to say that he had been playing around with a related shape that seemed to be behaving strangely,” Kaplan says. “Yoshiaki Araki, a Japanese mathematician and well-known artist whose work is in the spirit of MC Escher, had posted pictures of Tile(1,1) that got Dave interested in looking at it further.”

Yoshiaki posted an intriguing question on Twitter: “An aperiodic turtle tessellation based on new aperiodic monotile Tile(1, 1.1). In the tiling, it is said that around 12.7% of tiles are reflected. The green one is an instance. One more reflected turtle is hidden in the tiling. Who is the reflected?'”

“Yoshi had turned Tile(1,1) into turtles and it’s a bit hard to see the other reflected turtle in that picture. But it got Dave curious. What if we tile with this shape but without reflections? As he did that, Dave found that he could build tilings progressively outward in a pattern that didn’t stop and didn’t repeat.”

But then this shape came with a different quibble. As Kaplan explains, if you use reflections of Tile(1,1) the pattern does repeat. In other words, it’s periodic. But if Tile(1,1) is modified by replacing its straight edges with curves, it becomes a vampire einstein—a single shape that without reflection tiles the infinite plane in a pattern that can never be made to repeat.

The obvious question for mathematicians and tiling enthusiasts is what’s next?

“We can pose many variations of the problem,” Kaplan says. “The most interesting, for me at least, is whether this can be done in 3D. It would be nice to have a shape that repeats non-periodically in three dimensions. Such constructions are much harder to visualize, but computationally it’s not that much more difficult to prove should we be so lucky as to find a three-dimensional shape—a polyform—that like the hat tiles only aperiodically.”

“Tiling theory as a branch of mathematics is beautiful, tangible, and has a lot of fascinating problems to be solved. There’s no shortage of follow-up work to be done.”

Hatfest, a celebration of the discovery of “the Hat,” will be taking place at the University of Oxford’s Mathematical Institute from July 20 to 21. The event’s first day will feature talks and workshops on tiling aimed at a lay audience, while the second will feature presentations aimed at a broad audience of physicists and mathematicians.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Joe Petrik, University of Waterloo

左側的 14 邊形 Tile(1,1) 被稱為弱手性非週期單瓦 — 換句話說，如果禁止混合未反射和反射瓦片的鋪砌，則它僅以非週期性方式鋪砌。 然而，透過修改其邊緣，如中心和右側所示，可以創建稱為「幽靈」的嚴格手性非週期單瓦，這些單瓦僅允許非週期性鋪砌。 圖片來源：滑鐵盧大學

最近，一個由四人組成的國際團隊（包括 Cheriton 電腦科學學院教授 Craig Kaplan 博士）發現了一種單一形狀，可以鋪滿平面（一個無限的二維表面），其圖案永遠無法重複。

這一發現讓數學家、鋪砌愛好者和公眾都為之著迷。

這種形狀，一個 13 邊形，他們稱之為「帽子」，數學家稱之為非週期單瓦或「愛因斯坦」，這個德語詞的意思是「一塊石頭」。

但該團隊最近的發現再次提高了標準。 他們發現了另一種與第一種形狀相關的形狀，它符合更嚴格的定義。 這種被稱為「幽靈」的新形狀，以一種永遠不會重複的模式鋪滿平面，而無需使用該形狀的鏡像。 因此，它也被稱為「吸血鬼愛因斯坦」— 一種無需反射即可非週期性鋪砌的形狀。

「我們的第一篇論文解決了愛因斯坦問題，但由於該形狀需要反射才能進行非週期性鋪砌，因此人們提出了一個合理的問題：是否有一種形狀可以像帽子一樣，但無需反射，」Kaplan 解釋說。 「我們很幸運地找到了一種形狀，它不僅解決了這個子問題，而且在第一篇論文發表後不久就解決了它。」

對於數學家來說，帽子及其鏡像是一個單一的形狀，但在物理世界中，左手和右手的形狀表現可能不同。 例如，你不能在左手上戴右手手套。

「如果你用帽子形狀的瓷磚以非週期性方式鋪砌一個大型浴室地板，這些瓷磚的一面已經上釉，那麼你需要帽子和帽子的鏡像，」Kaplan 說。

但並非這種吹毛求疵促成了最近的發現。

吸血鬼愛因斯坦的發現始於 David Smith 的沉思，他是一位退休的印刷技術員，也是一位來自英國約克郡的自稱形狀愛好者，幾個月前他的好奇心促成了最初的愛因斯坦發現。

「在我們的帽子論文上線幾天后，Dave 給我們發了一封電子郵件，說他一直在玩一種相關的形狀，這種形狀的表現似乎很奇怪，」Kaplan 說。 「日本數學家兼著名藝術家 Yoshiaki Araki 的作品具有 MC Escher 的精神，他發布了 Tile(1,1) 的圖片，這引起了 Dave 的興趣，並進一步研究它。」

Yoshiaki 在 Twitter 上發布了一個有趣的問題：「基於新的非週期單瓦 Tile(1, 1.1) 的非週期海龜鑲嵌。在鑲嵌中，據說大約 12.7% 的瓷磚被反射。綠色的是一個例子。在鑲嵌中還隱藏著另一隻被反射的海龜。誰是被反射的？」

「Yoshi 將 Tile(1,1) 變成了海龜，很難在圖片中看到另一隻被反射的海龜。但這引起了 Dave 的好奇。如果我們用這種形狀鋪砌，但不進行反射呢？當他這樣做時，Dave 發現他可以逐步向外構建鋪砌，其模式不會停止也不會重複。」

但這種形狀帶有不同的吹毛求疵。 正如 Kaplan 解釋的那樣，如果你使用 Tile(1,1) 的反射，則圖案會重複。 換句話說，它是週期性的。 但是，如果通過用曲線替換其直線邊緣來修改 Tile(1,1)，它就會變成吸血鬼愛因斯坦 — 一種單一形狀，無需反射即可在永不重複的模式中鋪砌無限平面。

對於數學家和鋪砌愛好者來說，顯而易見的問題是下一步是什麼？

「我們可以提出這個問題的許多變體，」Kaplan 說。 「至少對我來說，最有趣的是這是否可以在 3D 中完成。如果有一種形狀可以在三個維度中非週期性地重複，那就太好了。這樣的結構很難視覺化，但如果我們有幸找到一種三維形狀（一種多形體），像帽子一樣僅以非週期性方式鋪砌，那麼在計算上證明它並不會困難得多。」

「作為數學的一個分支，鋪砌理論是美麗的、有形的，並且有很多有趣的問題需要解決。不乏後續工作要做。」

Hatfest，一個慶祝「帽子」發現的活動，將於 7 月 20 日至 21 日在牛津大學數學研究所舉行。 該活動的第一天將以針對普通觀眾的關於鋪砌的講座和研討會為特色，而第二天將以針對廣泛的物理學家和數學家觀眾的演示為特色。

如需更多此類見解，請登入我們的網站 https://international-maths-challenge.com

本文的作者為滑鐵盧大學的 Joe Petrik

Generations of scientists have compared the universe to a giant, complex game, raising questions about who is doing the playing – and what it would mean to win.

If the universe is a game, then who’s playing it?

The following is an extract from our Lost in Space-Time newsletter. Each month, we hand over the keyboard to a physicist or mathematician to tell you about fascinating ideas from their corner of the universe. You can sign up for Lost in Space-Time for free here.

Is the universe a game? Famed physicist Richard Feynman certainly thought so: “‘The world’ is something like a great chess game being played by the gods, and we are observers of the game.” As we observe, it is our task as scientists to try to work out the rules of the game.

The 17th-century mathematician Gottfried Wilhelm Leibniz also looked on the universe as a game and even funded the foundation of an academy in Berlin dedicated to the study of games: “I strongly approve of the study of games of reason not for their own sake but because they help us to perfect the art of thinking.”

Our species loves playing games, not just as kids but into adulthood. It is believed to have been an important part of evolutionary development – so much so that the cultural theorist Johan Huizinga proposed we should be called Homo ludens, the playing species, rather than Homo sapiens. Some have suggested that once we realised that the universe is controlled by rules, we started developing games as a way to experiment with the consequences of these rules.

Take, for example, one of the very first board games that we created. The Royal Game of Ur dates back to around 2500 BC and was found in the Sumerian city of Ur, part of Mesopotamia. Tetrahedral-shaped dice are used to race five pieces belonging to each player down a shared sequence of 12 squares. One interpretation of the game is that the 12 squares represent the 12 constellations of the zodiac that form a fixed background to the night sky and the five pieces correspond to the five visible planets that the ancients observed moving through the night sky.

But does the universe itself qualify as a game? Defining what actually constitutes a game has been a subject of heated debate. Logician Ludwig Wittgenstein believed that words could not be pinned down by a dictionary definition and only gained their meaning through the way they were used, in a process he called the “language game”. An example of a word that he believed only got its meaning through use rather than definition was “game”. Every time you try to define the word “game”, you wind up including some things that aren’t games and excluding others you meant to include.

Other philosophers have been less defeatist and have tried to identify the qualities that define a game. Everyone, including Wittgenstein, agrees that one common facet of all games is that they are defined by rules. These rules control what you can or can’t do in the game. It is for this reason that as soon as we understood that the universe is controlled by rules that bound its evolution, the idea of the universe as a game took hold.

In his book Man, Play and Games, theorist Roger Caillois proposed five other key traits that define a game: uncertainty, unproductiveness, separateness, imagination and freedom. So how does the universe match up to these other characteristics?

The role of uncertainty is interesting. We enter a game because there is a chance either side will win – if we know in advance how the game will end, it loses all its power. That is why ensuring ongoing uncertainty for as long as possible is a key component in game design.

Polymath Pierre-Simon Laplace famously declared that Isaac Newton’s identification of the laws of motion had removed all uncertainty from the game of the universe: “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past could be present before its eyes.”

Solved games suffer the same fate. Connect 4 is a solved game in the sense that we now know an algorithm that will always guarantee the first player a win. With perfect play, there is no uncertainty. That is why games of pure strategy sometimes suffer – if one player is much better than their opponent then there is little uncertainty in the outcome. Donald Trump against Garry Kasparov in a game of chess will not be an interesting game.

The revelations of the 20th century, however, have reintroduced the idea of uncertainty back into the rules of the universe. Quantum physics asserts that the outcome of an experiment is not predetermined by its current state. The pieces in the game might head in multiple different directions according to the collapse of the wave function. Despite what Albert Einstein believed, it appears that God is playing a game with dice.

Even if the game were deterministic, the mathematics of chaos theory also implies that players and observers will not be able to know the present state of the game in complete detail and small differences in the current state can result in very different outcomes.

That a game should be unproductive is an interesting quality. If we play a game for money or to teach us something, Caillois believed that the game had become work: a game is “an occasion of pure waste: waste of time, energy, ingenuity, skill”. Unfortunately, unless you believe in some higher power, all evidence points to the ultimate purposelessness of the universe. The universe is not there for a reason. It just is.

The other three qualities that Caillois outlines perhaps apply less to the universe but describe a game as something distinct from the universe, though running parallel to it. A game is separate – it operates outside normal time and space. A game has its own demarcated space in which it is played within a set time limit. It has its own beginning and its own end. A game is a timeout from our universe. It is an escape to a parallel universe.

The fact that a game should have an end is also interesting. There is the concept of an infinite game that philosopher James P. Carse introduced in his book Finite and Infinite Games. You don’t aim to win an infinite game. Winning terminates the game and therefore makes it finite. Instead, the player of the infinite game is tasked with perpetuating the game – making sure it never finishes. Carse concludes his book with the rather cryptic statement, “There is but one infinite game.” One realises that he is referring to the fact that we are all players in the infinite game that is playing out around us, the infinite game that is the universe. Although current physics does posit a final move: the heat death of the universe means that this universe might have an endgame that we can do nothing to avoid.

Caillois’s quality of imagination refers to the idea that games are make-believe. A game consists of creating a second reality that runs in parallel with real life. It is a fictional universe that the players voluntarily summon up independent of the stern reality of the physical universe we are part of.

Finally, Caillois believes that a game demands freedom. Anyone who is forced to play a game is working rather than playing. A game, therefore, connects with another important aspect of human consciousness: our free will.

This raises a question: if the universe is a game, who is it that is playing and what will it mean to win? Are we just pawns in this game rather than players? Some have speculated that our universe is actually a huge simulation. Someone has programmed the rules, input some starting data and has let the simulation run. This is why John Conway’s Game of Life feels closest to the sort of game that the universe might be. In Conway’s game, pixels on an infinite grid are born, live and die according to their environment and the rules of the game. Conway’s success was in creating a set of rules that gave rise to such interesting complexity.

If the universe is a game, then it feels like we too lucked out to find ourselves part of a game that has the perfect balance of simplicity and complexity, chance and strategy, drama and jeopardy to make it interesting. Even when we discover the rules of the game, it promises to be a fascinating match right up to the moment it reaches its endgame.

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

*Credit for article given to Marcus Du Sautoy*

一代又一代的科學家將宇宙比作一個巨大而複雜的遊戲，引發了關於誰在玩這個遊戲，以及獲勝意味著什麼的問題。

如果宇宙是場遊戲，那麼誰在玩它？

以下是我們《迷失於時空》電子報的摘錄。每個月，我們都會將鍵盤交給一位物理學家或數學家，向您講述他們宇宙角落中引人入勝的想法。您可以在此免費註冊《迷失於時空》。

宇宙是場遊戲嗎？著名物理學家理查·費曼當然這麼認為：「『世界』就像一場由眾神進行的偉大西洋棋遊戲，而我們是遊戲的觀察者。」當我們觀察時，我們作為科學家的任務是試圖找出遊戲規則。

17 世紀數學家戈特弗里德·威廉·萊布尼茨也將宇宙視為一場遊戲，甚至資助了柏林一所專門研究遊戲的學院的成立：「我非常贊成研究理性遊戲，不是為了遊戲本身，而是因為它們有助於我們完善思考的藝術。」

我們人類喜歡玩遊戲，不僅僅是孩子，成年後也是如此。據信，遊戲是進化發展的重要組成部分——以至於文化理論家約翰·赫伊津哈（Johan Huizinga）提出，我們應該被稱為 Homo ludens，即遊戲物種，而不是 Homo sapiens。有人認為，一旦我們意識到宇宙是由規則控制的，我們就開始開發遊戲，以此來試驗這些規則的後果。

以我們創造的最早的棋盤遊戲之一為例。烏爾王室遊戲可以追溯到西元前 2500 年左右，在美索不達米亞的蘇美爾城市烏爾被發現。四面體形狀的骰子用於競賽，每個玩家的五個棋子沿著共享的 12 個方格的序列前進。對該遊戲的一種解釋是，這 12 個方格代表黃道帶的 12 個星座，它們構成了夜空的固定背景，而五個棋子對應於古代人觀察到的在夜空中移動的五顆可見行星。

但是宇宙本身是否符合遊戲的資格？定義什麼構成遊戲一直是激烈爭論的主題。邏輯學家路德維希·維根斯坦認為，詞語不能被字典定義所束縛，只能通過它們的使用方式來獲得意義，他稱之為「語言遊戲」。他認為，只有通過使用而不是定義才能獲得意義的一個詞的例子是「遊戲」。每次你試圖定義「遊戲」這個詞時，你最終會包括一些不是遊戲的東西，並排除其他你想要包括的東西。

其他哲學家則不那麼悲觀，並試圖確定定義遊戲的品質。包括維根斯坦在內的每個人都同意，所有遊戲的一個共同點是它們由規則定義。這些規則控制著你在遊戲中可以做什麼或不能做什麼。正是因為這個原因，一旦我們理解了宇宙是由約束其演化的規則控制的，宇宙作為遊戲的想法就紮根了。

理論家羅傑·凱約瓦（Roger Caillois）在他的著作《人、遊戲和玩耍》中提出了定義遊戲的其他五個關鍵特徵：不確定性、非生產性、分離性、想像力和自由。那麼，宇宙在多大程度上符合這些其他特徵呢？

不確定性的作用很有趣。我們進入遊戲是因為任何一方都有可能獲勝——如果我們事先知道遊戲將如何結束，它就會失去所有的力量。這就是為什麼確保盡可能長時間的持續不確定性是遊戲設計中的一個關鍵組成部分。

博學家皮埃爾-西蒙·拉普拉斯（Pierre-Simon Laplace）曾著名地宣稱，艾薩克·牛頓（Isaac Newton）對運動定律的識別消除了宇宙遊戲中的所有不確定性：「我們可以將宇宙的現在狀態視為其過去的結果和其未來的起因。一個在某一時刻知道所有推動自然運動的力量，以及組成自然的所有物體的所有位置的智力，如果這個智力也足夠廣闊，可以將這些數據提交給分析，它將在一個單一的公式中包含宇宙中最大物體的運動和最小原子的運動；對於這樣一個智力來說，沒有什麼是不確定的，未來就像過去一樣可以呈現在它的眼前。」

已解決的遊戲遭受同樣的命運。四子棋是一種已解決的遊戲，因為我們現在知道一種演算法，它總是能保證先手玩家獲勝。在完美的遊戲中，沒有不確定性。這就是為什麼純粹的策略遊戲有時會受到影響——如果一個玩家比他們的對手好得多，那麼結果幾乎沒有不確定性。唐納德·川普（Donald Trump）對陣加里·卡斯帕羅夫（Garry Kasparov）的西洋棋比賽將不會是一場有趣的比賽。

然而，20 世紀的啟示重新將不確定性的概念引入了宇宙的規則中。量子物理學斷言，實驗的結果並非由其當前狀態預先決定。遊戲中的棋子可能會根據波函數的崩潰朝多個不同的方向前進。儘管阿爾伯特·愛因斯坦（Albert Einstein）相信什麼，但上帝似乎正在擲骰子玩遊戲。

即使遊戲是確定性的，混沌理論的數學也暗示著玩家和觀察者將無法完全詳細地了解遊戲的當前狀態，並且當前狀態的微小差異可能導致非常不同的結果。

遊戲應該是非生產性的，這是一個有趣的品質。如果我們為了錢或為了教我們一些東西而玩遊戲，凱約瓦認為遊戲已經變成了工作：遊戲是「純粹浪費的場合：浪費時間、精力、才智、技能」。不幸的是，除非你相信某種更高的力量，否則所有證據都指向宇宙的最終無目的性。宇宙不是為了某個原因而存在的。它只是存在。

凱約瓦概述的其他三個品質可能較少適用於宇宙，但將遊戲描述為與宇宙不同的東西，儘管與宇宙平行運行。遊戲是分離的——它在正常的時間和空間之外運行。遊戲有其自己劃定的空間，在設定的時間限制內進行。它有自己的開始和自己的結束。遊戲是我們宇宙的暫停。它是逃往平行宇宙的途徑。

遊戲應該有一個結束也是很有趣的。哲學家詹姆斯·P·卡斯（James P. Carse）在他的著作《有限與無限遊戲》中介紹了無限遊戲的概念。你的目標不是贏得無限遊戲。獲勝會終止遊戲，因此使其成為有限的。相反，無限遊戲的玩家的任務是延續遊戲——確保它永遠不會結束。卡斯以相當隱晦的陳述結束了他的書：「只有一個無限遊戲。」人們意識到他指的是我們都是在我們周圍進行的無限遊戲的玩家，這個無限遊戲就是宇宙。儘管目前的物理學確實提出了一個最終的舉動：宇宙的熱寂意味著這個宇宙可能有一個我們無能為力的殘局。

凱約瓦的想像力品質指的是遊戲是虛構的想法。遊戲包括創造一個與現實生活平行運行的第二現實。這是一個虛構的宇宙，玩家自願召喚它，獨立於我們所屬的物理宇宙的嚴酷現實。

最後，凱約瓦認為遊戲需要自由。任何被迫玩遊戲的人都是在工作而不是在玩耍。因此，遊戲與人類意識的另一個重要方面聯繫在一起：我們的自由意志。

這提出了一個問題：如果宇宙是場遊戲，那麼誰在玩它，獲勝意味著什麼？我們只是這個遊戲中的棋子而不是玩家嗎？有人推測我們的宇宙實際上是一個巨大的模擬。有人編寫了規則，輸入了一些起始數據，並讓模擬運行。這就是為什麼約翰·康威（John Conway）的生命遊戲感覺最接近宇宙可能存在的遊戲類型。在康威的遊戲中，無限網格上的像素根據它們的環境和遊戲規則而誕生、生存和死亡。康威的成功在於創造了一套規則，這些規則產生了如此有趣的複雜性。

如果宇宙是場遊戲，那麼我們也感到幸運，發現自己是遊戲的一部分，這個遊戲在簡單性和複雜性、機遇和策略、戲劇性和危險性之間取得了完美的平衡，使其變得有趣。即使我們發現了遊戲規則，它也承諾在達到殘局的那一刻之前，都將是一場引人入勝的比賽。

如需更多此類見解，請登錄 www.international-maths-challenge.com。

*文章作者：Marcus Du Sautoy*

“We wanted to investigate how second pillar pension funds react to financial crises and how to protect them from the crises,” says Kaunas University of Technology (KTU) professor Dr. Audrius Kabašinskas, who, together with his team, discovered a way to achieve this goal. The discovery in question is the development of stress tests for pension funds. Lithuanian researchers were the first in the world to come up with such an adaptation of the stress tests.

Stress tests are usually carried out on banks or other financial institutions to allow market regulators to determine and assess their ability to withstand adverse economic conditions.

According to the professor at KTU Faculty of Mathematics and Natural Sciences, this innovative pension fund stress testing approach will benefit both regulators and pension fund managers.

“Making sure your pension fund is resilient to harsh financial market conditions will help you sleep better, save more, and have increased trust in your funds and the pension system itself,” Kabašinskas adds.

Results based on two major crises

First, the study needed to collect data from previous periods. “Two major events that shocked the whole world—COVID-19 and the first year of Russian invasion of Ukraine—just happened to occur during the project. This allowed us to gather a lot of relevant information and data on changes in the performance of pension funds,” says Kabašinskas.

The Hidden Markov Model (HMM), which, according to a professor at KTU Department of Mathematical Modelling, is quite simple in its principle of operation, helped to forecast future market conditions in this study.

The paper is published in the journal Annals of Operations Research.

“The observation of air temperature could be an analogy for it. All year round, without looking at the calendar, we observe the temperature outside and, based on the temperature level, we decide what time of the year it is. Of course, 15 degrees can occur in winter and sometimes it snows in May but these are random events. The state of the next day depends only on today,” he explains vividly.

According to the KTU researcher, this describes the idea of the Hidden Markov Model: by observing the changes in value, one can judge the state of global markets and try to forecast the future.

“In our study, we observed two well-known investment funds from 2019 to 2022. Collected information helped us identify that global markets at any given moment are in one of four states: no shock regime, a state of shock in stock markets, a state of shock in bond markets, and a state of global financial shock—a global crisis,” says Kabašinskas.

Using certain methods, the research team led by a professor Miloš Kopa representing KTU and Charles University in Prague found that these periods were aligned with the global events in question. Once the transition probabilities between the states were identified, it was possible to link the data of pension funds to these periods and simulate the future evolution of the pension funds’ value.

That’s where the innovation of stress testing came in. The purpose of this test is to determine whether a particular pension fund can deliver positive growth in the future when faced with a shock in the financial markets.

“In our study, we applied several scenarios, extending financial crises and modeling the evolution of fund values over the next 5 years,” says a KTU researcher.

This methodology can be applied not only to pension funds but also to other investments.

Example of Lithuanian pension funds

The research and the new stress tests were carried out on Lithuanian pension funds.

Kabašinskas says that the study revealed several interesting things. Firstly, on average, Lithuanian second pillar pension funds can withstand crises that are twice as long.

“However, the results show that some Lithuanian funds struggle to cope with inflation, while others, the most conservative funds for citizens who are likely to retire within next few years or who have already retired, are very slow in recovering after negative shocks,” adds the KTU expert.

This can be explained by regulatory aspects and the related investment strategy, as stock markets recover several times faster than bond markets, and the above-mentioned funds invest more than 90% in bonds and other less risky instruments.

A complementary study has also been carried out to show how pension funds should change their investment strategy to avoid the drastic negative consequences of various financial crises and shocks.

“Funds that invest heavily in stocks and other risky instruments should increase the number of risk-free instruments slightly, up to 10%, before or after the financial crisis hits. Meanwhile, funds investing mainly in bonds should increase the number of stocks in their holdings. In both cases, the end of the crisis should be followed by a slow return to the typical strategy,” advises a mathematician.

Although the survey did not aim to increase people’s confidence in pension funds, the results showed that Lithuania’s second pillar pension funds are resilient to crisis and are worth trust. Historically they have delivered long-term growth, some have even outperformed inflation and price increases.

“Although short-term changes can be drastic, long-term growth is clearly visible,” says KTU professor Dr. Kabašinskas. “Lithuania, by the way, has a better system than many European countries,” he adds.

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

Finding bats is hard. They are small, fast and they primarily fly at night.  But our new research could improve the way conservationists find bat roosts. We’ve developed a new algorithm that significantly reduces the area that needs to be searched, which could save time and cut labour cost.

Of course, you may wonder why we would want to find bats in the first place. But these flying mammals are natural pest controllers and pollinators, and they help disperse seeds. So they are extremely useful in contributing to the health of our environment.

Despite their importance though, bat habitats are threatened by human activities such as increased lighting, noise and land use. To ensure that we can study and enhance the health of our bat population, we need to locate their roosts. But finding bat roosts is a bit like finding a needle in a haystack.

Our previous work measured and modelled the motion of greater horseshoe bats in flight. Having such a model means we can predict where bats will be, depending on their roost position. But the position of the roost is something we often don’t know.

Our new research combines our previous mathematical model of bat motion with data gathered from acoustic recorders known as “bat detectors”. These bat detectors are placed around the environment and left there for several nights.

Seeing with sound

Bats use echolocation, which allows them to “see with sound” when they’re flying. If these ultrasonic calls are made within ten to 15 metres of a bat detector, the device is triggered to make a recording, providing an accurate record of where and when a bat was present.

The sound recordings also provide clues about the identity of the species. Greater horseshoe bats make a very distinctive “warbling” call at almost exactly 82kHz in frequency, so we can easily tell whether the species is present or not.

Assuming that a bat detector’s batteries last for a few nights, its memory card is not full, and the units are not stolen or vandalised, then we can use the bat call data to generate a map that shows the proportion of bat calls at each detector location.

Our model can also be used to predict the proportion of bat calls based on a given roost location. So, we split the environment up into a grid and simulate bats flying from each grid square. The grid square, or squares, whose simulations best reproduce the bat detector data will then be the most likely locations of the roost.

This simple algorithm can then be applied to whole terrains, meaning that we can create a map of likely roost locations. Cutting out the regions that are least likely to contain the roost can mean we shrink the search space to less than 1% of the initially surveyed area. Simplifying the process of finding bat roosts allows more of an ecologist’s time to be spent on conservation projects, rather than laborious searching.

In 2022, we developed an app that uses publicly available data to predict bat flight lines. At the moment the app can help ecologists, developers or local authority planners, know how the environment is used by bats. However, it needs a roost location to be specified first, and this information is not always known. Our new research removes this barrier, making the app easier to use.

Our work offers a way of identifying likely roost locations. These estimates can then be verified either by directly observing particular features, or by capturing bats at a nearby location and following them back home, using radiotracking.

Over the past two decades, bat detectors have gone from simple hand-held machines to high-performance devices that can collect data for days at a time. Yet they are usually deployed only to identify bat species. We have shown they can be used to identify the areas most likely to contain bat roosts, uncovering critical information about these most secretive of animals.

We hope that this will provide further tools for ecologists to optimise the initial microphone detector locations, thereby providing a holistic way of detecting bat roosts.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Thomas Woolley and Fiona Mathews, The Conversation

Direct reciprocity facilitates cooperation in repeated social interactions. Traditional models suggest that individuals learn to adopt conditionally cooperative strategies if they have multiple encounters with their partner. However, most existing models make rather strong assumptions about how individuals decide to keep or change their strategies. They assume individuals make these decisions based on a strategy’s average performance. This in turn suggests that individuals would remember their exact payoffs against everyone else.

In a recent study, researchers from the Max Planck Institute for Evolutionary Biology, the School of Data Science and Society, and the Department of Mathematics at the University of North Carolina at Chapel Hill examine the effects of realistic memory constraints. They find that cooperation can evolve even with minimal memory capacities. The research is published in the journal Proceedings of the Royal Society B: Biological Sciences.

Direct reciprocity is based on repeated interactions between two individuals. This concept, often described as “you scratch my back, I’ll scratch yours,” has proven to be a pivotal mechanism in maintaining cooperation within groups or societies.

While models of direct reciprocity have deepened our understanding of cooperation, they frequently make strong assumptions about individuals’ memory and decision-making processes. For example, when strategies are updated through social learning, it is commonly assumed that individuals compare their average payoffs.

This would require them to compute (or remember) their payoffs against everyone else in the population. To understand how more realistic constraints influence direct reciprocity, the current study considers the evolution of conditional behaviours when individuals learn based on more recent experiences.

Two extreme scenarios

This study first compares the classical modeling approach with another extreme approach. In the classical approach, individuals update their strategies based on their expected payoffs, considering every single interaction with each member of the population (perfect memory). Conversely, the opposite extreme is considering only the very last interaction (limited memory).

Comparing these two scenarios shows that individuals with limited payoff memory tend to adopt less generous strategies. They are less forgiving when someone defects against them. Yet, moderate levels of cooperation can still evolve.

Intermediate cases

The study also considers intermediate cases, where individuals consider their last two or three or four recent experiences. The results show that cooperation rates quickly approach the levels observed under perfect payoff memory.

Overall, this study contributes to a wider literature that explores which kinds of cognitive capacities are required for reciprocal altruism to be feasible. While more memory is always favourable, reciprocal cooperation can already be sustained if individuals have a record of two or three past outcomes.

This work’s results have been derived entirely within a theoretical model. The authors feel that such studies are crucial for making model-informed deductions about reciprocity in natural systems.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Michael Hesse, Max Planck Society

Bob Moses, who helped register Black residents to vote in Mississippi during the Civil Rights Movement, believed civil rights went beyond the ballot box. To Moses, who was a teacher as well as an activist, math literacy is a civil right: a requirement to earning a living wage in modern society. In 1982, he founded the Algebra Project to ensure that “students at the bottom get the math literacy they need.”

As a researcher who studies ways to improve the math experiences of students, WEbelieve a new approach that expands access to algebra may help more students get the math literacy Moses, who died in 2021, viewed as so important. It’s a goal districts have long been struggling to meet.

Efforts to improve student achievement in algebra have been taking place for decades. Unfortunately, the math pipeline in the United States is fraught with persistent opportunity gaps. According to the Nation’s Report Card – a congressionally mandated project administered by the Department of Education – in 2022 only 29% of U.S. fourth graders and 20% of U.S. eighth graders were proficient in math. Low-income students, students of color and multilingual learners, who tend to have lower scoreson math assessments, often do not have the same access as others to qualified teachers, high-quality curriculum and well-resourced classrooms.

A new approach

The Dallas Independent School District – or Dallas ISD – is gaining national attention for increasing opportunities to learn by raising expectations for all students. Following in the footsteps of more than 60 districts in the state of Washington, in 2019 the Dallas ISD implemented an innovative approach of having students be automatically enrolled rather than opt in to honours math in middle school.

Under an opt-in policy, students need a parent or teacher recommendation to take honours math in middle school and Algebra 1 in eighth grade. That policy led both to low enrollment and very little diversity in honours math. Some parents, especially those who are Black or Latino, were not aware how to enroll their students in advanced classes due to a lack of communication in many districts.

In addition, implicit bias, which exists in all demographic groups, may influence teachers’ perceptions of the behaviour and academic potential of students, and therefore their subsequent recommendations. Public school teachers in the U.S. are far less racially and ethnically diverse than the students they serve.

Dallas ISD’s policy overhaul aimed to foster inclusivity and bridge educational gaps among students. Through this initiative, every middle school student, regardless of background, was enrolled in honours math, the pathway that leads to taking Algebra 1 in eighth grade, unless they opted out.

Flipping the switch from opt-in to opt-out led to a dramatic increase in the number of Black and Latino learners, who constitute the majority of Dallas students. And the district’s overall math scores remained steady. About 60% of Dallas ISD eighth graders are now taking Algebra 1, triple the prior level. Moreover, more than 90% are passing the state exam.

Civil rights activist Bob Moses believed math literacy was critical for students to be able to make a living. Robert Elfstrom/Villon Films via Getty Images

Efforts spread

Other cities are taking notice of the effects of Dallas ISD’s shifting policy. The San Francisco Unified School District, for example, announced plans in February 2024 to implement Algebra 1 in eighth grade in all schools by the 2026-27 school year.

In fall 2024, the district will pilot three programs to offer Algebra 1 in eighth grade. The pilots range from an opt-out program for all eighth graders – with extra support for students who are not proficient – to a program that automatically enrolls proficient students in Algebra 1, offered as an extra math class during the school day. Students who are not proficient can choose to opt in.

Nationwide, however, districts that enroll all students in Algebra 1 and allow them to opt out are still in the minority. And some stopped offering eighth grade Algebra 1 entirely, leaving students with only pre-algebra classes. Cambridge, Massachusetts – the city in which Bob Moses founded the Algebra Project – is among them.

Equity concerns linger

Between 2017 and 2019, district leaders in the Cambridge Public Schools phased out the practice of placing middle school students into “accelerated” or “grade-level” math classes. Few middle schools in the district now offer Algebra 1 in eighth grade.

The policy shift, designed to improve overall educational outcomes, was driven by concerns over significant racial disparities in advanced math enrollment in high school. Completion of Algebra 1 in eighth grade allows students to climb the math ladder to more difficult classes, like calculus, in high school. In Cambridge, the students who took eighth grade Algebra 1 were primarily white and Asian; Black and Latino students enrolled, for the most part, in grade-level math.

Some families and educators contend that the district’s decision made access to advanced math classes even more inequitable. Now, advanced math in high school is more likely to be restricted to students whose parents can afford to help them prepare with private lessons, after-school programs or private schooling, they said.

While the district has tried to improve access to advanced math in high school by offering a free online summer program for incoming ninth graders, achievement gaps have remained persistently wide.

Perhaps striking a balance between top-down policy and bottom-up support will help schools across the U.S. realize the vision Moses dreamed of in 1982 when he founded the Algebra Project: “That in the 21st century every child has a civil right to secure math literacy – the ability to read, write and reason with the symbol systems of mathematics.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Liza Bondurant, The Conversation