Maths Ed Ideas: On Learning When You're Not Looking #1

Whether or not it is possible to be aware that we don't know something when we are not looking for it, we become only too aware when we find it — too aware that is of our not knowing, or of our neglect in having lost sight of the sense of wonder that comes with finding something out for the first time.

Finding things out when we are not looking has resonance for us, emotionally, intellectually, and formatively, and particularly so I think for teachers, because it is a rediscovery of sorts in ourselves of a sense of naïvety that we see — rightly or wrongly — in our students, a naïvety moreover imbued with the pedagogical promise that it can be made something of, an inquisitiveness in other words and capacity to be surprised that we justly idealise and cultivate in our students. The resonance of finding something out when we are not looking, of going places we weren't planning or perhaps prepared to go, is, in short, and to put it another way, a rediscovery of our own curiosity. And the more we are subject to this resonance, the more we want more of it, the more we want to share it.

To this end, I offer here the first of an occasional series of posts in which I share a few little somethings that have made me wonder how I have got this far in life without either knowing, realising or appreciating more. They may be titbits of trivia with a mathematical bent that I have stumbled across, or something whose resonance for others has vicariously re-awoken in me the fascination they once first instilled in me. But most importantly, they have the pedagogical, educative and aspirational capital to exploit in support of our students' mathematical maturation, and to encourage students to begin to see themselves as young mathematicians in the making.

Click to jump to a little something about:

Blockbusters and the Game of Hex
(Click here to go back to the menu)

The TV game show Blockbusters, created by prolific puzzle writer Steve Ryan [1], and still often used as a conceit to 'quiz' students in classrooms, was based on the Game of Hex invented by John Nash, the mathematician and winner of the Nobel Prize [2] for economics for his work on game theory, and whose life story was dramatised in the 2004 Oscar winning film 'A Beautiful Mind'. [Heard from Tim Harford at the start of 'Tell Me Something I Don't Know' podcast episode 15.]

T
The game — Hex, that is, not Blockbusters — was actually first invented (or, arguably, discovered) in 1942 by Piet Hein, a Danish mathematician and polymath described by Martin Gardner as 'surely one of the most remarkable men in Denmark' (2008, p82), while he was working on the four-colour theorem. The game 'became enormously popular in Denmark under the name of Polygon' (ibid), before John Nash subsequently and independently re-discovered the game in 1948, at Princeton (this deleted scene from 'A Beautiful Mind' is an interesting watch). The game became popular amongst Princeton's mathematics graduate students who called it "Nash" or "John", the latter in affectionate homage to the hexagonal bathroom tiles that the students often played the game on. (For more about the game, its history and its mathematics, see this from Wolfram Mathworld, and this from MIT.)

Hex is played on a diamond-shaped board made up of hexagons. Two players take it in turns to mark a hexagon with their assumed colour, orange and white in the board illustrated above, and win by connecting their opposite sides of the board in an unbroken chain. The game is a 'determined' game, i.e. a game that can never end in a draw with the first player always being able to force a win on a standard board of any size, as proved reductio ad absurdum by Nash in his 'strategy stealing' argument, as described by Gardner (2008, p85) and exquisitely here by Alexander Bogomolny.

In blocking your opponent's path across the board, you will of course connect your sides of the board, and if you go first you will always have that 'extra' first mark to facilitate the block and thus complete your unbroken chain — in theory. But whilst Nash proved that on symmetrical boards of any size Hex is a 'first player' win (read an interesting discussion here), he did not show what the strategy is. Indeed, the first explicit winning strategy (on a 7×7 board) was described in 2002, and as of 2016, by using brute force search computer algorithms, Hex boards up to size 9×9 have been completely solved — but no further. As Bogomolny puts it, as 'simple as the game appears, no winning strategy has been discovered for bigger boards'.

This 4 minute video from Marc Chamberland of Tipping Point Math gives an excellent overview of the game and its significance.

To play the game with students (on a seven-by-seven board) this spreadsheet, by James Robinson from @ATMMathematics, and this lovely geogebra file from Steve Phelps both work well — through projection, perhaps on an interactive whiteboard. For more independent exploratory pencil and paper play, I provide an A4 version of the standard eleven-by-eleven board to download here.

As Gardner states, 'one of the best ways to learn the subtleties of Hex is to play the game on a field with a small number of hexagons' (2008, p84), building up gradually from a two-by-two board of four hexagons. So after some initial exploratory play perhaps on the seven-by-seven and/or eleven-by-eleven boards provided above, by considering the game on boards of gradually increasing size (available to download in various formats here), students will quickly see that — and appreciate how — the first player wins on the two-by-two and three-by-three boards, but that things get more complicated quickly from then on (and with opportunities therein for deeper learning). On the four-by-four board, for example, as the maestro explains, 'the first player is sure to win if he immediately occupies any one of the four cells [in the centre column of hexagons]. If he makes his opening play elsewhere, he can always be defeated. An opening play in [the two centre hexagons] insures a win on the fifth move; an opening play in [a hexagon above or below the two centre hexagons], a win on the sixth move.' Gardner continued, 'the standard 11-by-11 board introduces such an astronomical number of complications that a complete analysis seems beyond the range of human computation', (ibid, p84).

It is interesting also to consider Nash's proof with students, and then to vary the rules of the game and explore with students what happens to their underlying strategy if, for example, the idea of winning is switched so that you win by avoiding winning (i.e. by not connecting your sides). Or maybe by applying new rules, for example by having each player being allowed to swap positions once during each game.

The Isoperimetric Inequality
(Click here to go back to the menu)

The Isoperimetric Inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that:
\[A \le \frac{{{L^2}}}{{4\pi }}\]
and that equality holds if and only if the curve is a circle. [Heard (again) here thanks to @panlepan's nomination to @TeachFMaths 'Favourite Theorems and Conjectures' Twitter #MathOlympiad2018, and subsequently through perusing this by Robert Osserman in the November 1978 Bulletin of the American Mathematical Society]

F
Finding out some years ago that something I had independently and innocuously pondered upon as a youngster, that something I had intuitively understood as a schoolboy had a name, and that it had a name because it had a long history behind it [3], a history moreover of real, elegant and rigorous maths [4], had a profound effect on me — personally as a young learner, but more so I think professionally since I became a teacher.

So I was spiked by a desire to share the idea with students, or, moreover, to engineer a situation through which students may be able to pose and ponder the idea for themselves, without — to put it another way — giving it up up front. As such I sketched out this problem [5], as shown below, linking the perimeter to the area of a curved 'fidget-spinnery' shape of sorts not typically considered in students' mathematics lessons, and that aims thus to nudge students (with careful questioning) towards wondering whether it is possible for a shape with the same perimeter as a circle's to have a larger area than the circle.

T
The idea is that working through this 'fidget-spinnery' problem with students would grant them the intellectual space to essentially pose the Isoperimetric Problem themselves, inducing the resonance of learning-when-you're-not-looking that in turn provides us as teachers with the opportunity to celebrate and validate students' own problematising (a key characteristic of the mathematically maturing student), by framing it within — and revealing the — history of the problem.

In other words, it is about using the problem to support the ongoing cultivation of a pedagogically empathetic classroom that tries to answer questions students think to ask, or questions we are asking as teachers that students if not love at least like, as Steven Strogatz elucidates beautifully in this article for the American Mathematical Society.

O
Once the problem is 'out there' [6], as it were, students may begin by exploring the area (A) of rectangles with a fixed perimeter (P) — e.g. for P = 20, {1,1,9,9} A = 9... {2,2,8,8} A = 16... {3,3,7,7} A = 21... {4,4,6,6} A = 24... {5,5,5,5} A = 25 — before moving on to other shapes, ideally suggested by students themselves (validated or moderated by the teacher) and ultimately perhaps into how the area of an ellipse changes as we vary the lengths of its major (a) and minor (b) axes, whilst keeping the circumference constant and equal to the circumference of a circle (i.e. when a = b).

This provides the opportunity to reveal for students the perhaps surprising, exciting knowledge that whilst we know the area of an ellipse (πab), we have no exact formula for the cirumference of an ellipse, other, of course, than the sum of an infinite series (see this typically helpful consideration from mathsisfun.com). This in turn would provide us with the 'excuse' to share some of the mathematics of Ramanujan with students, through the use of his approximations for an ellipse's circumference, perhaps:

\[p \approx \pi \left[ {3\left( {a + b} \right) - \sqrt {\left( {3a + b} \right)\left( {a + 3b} \right)} } \right]\]

Or this:

\[p \approx \pi \left( {a + b} \right)\left[ {1 + \frac{{3{{\left( {\frac{{a - b}}{{a + b}}} \right)}^2}}}{{10 + \sqrt {4 - 3{{\left( {\frac{{a - b}}{{a + b}}} \right)}^2}} }}} \right]\]

In exploring then how the area of an ellipse changes as a and b vary until a = b, students find respective values for a and b for a constant circumference, meaning that in terms thus of the Key Stage 4 National Curriculum in England — Algebra Programme of Study, 'additional mathematical content... taught to more highly attaining pupils' — students will be 'find[ing] approximate solutions to equations numerically using iteration'. For example, for a circumference of 20, using Ramanujan's first formula above, when a = 1, b ≈ 4.739916655 and thus A ≈ 14.89088734. When a = 2, b ≈ 4.173401125 and thus A ≈ 26.22225263. When a = 3, b ≈ 3.361072807 and thus A ≈ 31.67736492. When a = b = 3.183098862 and thus A = 31.83098862.

Wang Tiles (or dominoes)
(Click here to go back to the menu)

I came across 'Wang Tiles' whilst absentmindedly browsing McSweeney's Internet Tendency (@mcsweeneys) and stumbled across Mark Amundsen's 'Tenth-Graders' Favorite Suggestive Terms from Geometry' from 2008.

The Chinese American mathematician, logician, and philosopher, Hao Wang first proposed a method for deciding an important case of David Hilbert's Entscheidungsproblem (the 'Decision Problem', a challenge posed by David Hilbert in 1928), in a 1961 paper. This method is known as ‘the domino problem’, and was popularised in Wang’s November 1965 Scientific American article ‘Games, Logic and Computers’.

A Wang tile (or domino) is a unit square whose edges are coloured (or otherwise marked with specific symbols) in various ways, and Wang's problem was to find a procedure for deciding whether an infinite plane can be covered with a set of such tiles so that the colours on adjacent edges match, and with no tile being rotated or reflected. Wang conjectured that if a set of tiles can cover the infinite plane, then there must be at least one arrangement in which they do so periodically (like wallpaper say), and thus that there would be a procedure for deciding whether a set of tiles can tile the infinite plane or not. Robert Berger, however, one of Wang's students showed in 1966 that there exists sets of Wang tiles that cover the plane, but only aperiodically [7], and constructed such a set of over 20,000 tiles (the illustration above shows such a set of 13 Wang tiles [8]). In 2015 Emmanuel Jeandel and Michael Rao proved 'that there are no aperiodic tile set with less than 11 Wang tiles, and that there is an aperiodic tile set with 11 Wang tiles and 4 colors.'

Wang Tiles are an interesting conceit for exploring ideas of symmetry, combinations and permutations, logic, and algorithmic thinking with students. One avenue of exploration could come from students creating their own tile sets and exploring how to tile grids of specific sizes. They could use Jeandel and Rao's astonishing 2015 finding to see what periodic tilings they could make (albeit on a finite plane!) with a set of less than 11 tiles less using 4 colours or less. They may want to start trivially and systematise their explorations.

Students may also find inspiration in the applications of Wang tiles, and particularly perhaps with reference to texturing in computer graphics. This excellent introduction to the use of Wang tiles in computer graphics from Miguel Cepero, along with this on Wang Tiles and aperiodical tiling by Graham Shawcross, could provide some ideas and inspiration for work with students on combinations. This from Alan Wolfe gives a succinct explanation about how to use Wang tiles to generate 'tile based graphics that don't look tiled at all', and this from Kevin Roberge is a fascinating account on using Wang Tiles for computation (e.g. 2 + 2 = 4). Wolfram's Demonstrations Project has this demonstration, a variation of the idea of Wang Tiles, to generate tiling patterns.

Notes, References & Links:

[1] See this, for example, by buzzerblog, a website dedicated to game show entertainment industry.

[2] Follow this link to watch John Nash receive his Nobel Prize in Economics in 1994

[3] See Viktor Blåsjö's excellent article in the June-July 2005 issue of the Mathematical Association of America monthly: 'The Evolution of the Isoperimetric Problem'.

[4] This by Alexander Bogomolny again is a superb elucidation of the theorem and inequality.

[5] Complete with a solution 'think through'.

[6] You may find this proof of the Isoperimetric Theorem, by Herman Gluck from the University of Pennsylvania, interesting / useful. And this from Warwick University's mathematics undergraduate handbook might be fun to share with students after work on the(ir) problem.

[7] This by Dylan Mohlo from Williams College Mathematics and Statistics, gives a nice narrative to 'The Undecidability of the Domino Problem and the Creation of the First Aperiodic Tiling'.

[8] Adapted from Culik (1996), where he proved an aperiodic set of 13 tiles with 5 colours.

Amundsen, M. (2008, 22 October). 'Tenth-Graders' Favorite Suggestive Terms from Geometry', McSweeney's Internet Tendency. Retrieved January 2018 from https://www.mcsweeneys.net/articles/tenth-graders-favorite-suggestive-terms-from-geometry

andStack (2011, January 20). A Beautiful Mind Deleted Scene - A New Game of Go [Video File]. Retrieved from https://youtu.be/pTZ3nn2Bge4

Berger, R. (1966). 'The Undecidability of the Domino Problem.' Memoirs of the American Mathematical Society, 66. Retrieved from https://bookstore.ams.org/memo-1-66/

Blåsjö, V. (2005). 'The Evolution Of...The Isoperimetric Problem', Mathematical Association of America monthly, 112, June-July 2005, pp526-566. Retrieved February 2018 from https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/blasjo526.pdf

Blockbusters (UK Game Show). (n.d.). In Wikipedia. Retrieved February 2018 from https://en.wikipedia.org/wiki/Blockbusters_(UK_game_show)

Bogomolny, A. (n.d.). 'Hex Can't End in a Draw', from Interactive Mathematics Miscellany and Puzzles, https://www.cut-the-knot.org/Curriculum/Games/HexTheory.shtml. Accessed February 2018

Bogomolny, A. (n.d.). 'Isoperimetric Theorem and Inequality', from Interactive Mathematics Miscellany and Puzzles, https://www.cut-the-knot.org/do_you_know/isoperimetric.shtml, Accessed February 2018.

Cepero, M. (2013, January 3). Introduction to Wang Tiles [Blog post]. Retrieved from http://procworld.blogspot.co.uk/2013/01/introduction-to-wang-tiles.html

Chamberland, M. [Tipping Point Math]. (2016, 11 Novmeber). The Joy of Hex Retrieved February 2018 from https://youtu.be/2MNalT1g3m8