Math Circle Session 9: Parity

Today’s session was inspired by a different math circle book: that by Anna Burago, which is targeted at slightly older children.

We started with 7 coins, and asked the children to place them so that 4 were heads up and three tails up. The rules of the game are as follows:

  • in each turn, you can pick any two coins of your choice
  • you must turn both those coins over

Your aim is to get all 7 coins facing heads up.

The task is impossible, but the children didn’t see so immediately. One child even thought he had found a solution, but was unable to repeat it. One child noticed that he could get them to all tails, but not to all heads. Very quickly, children of all ages noticed that “odds” and “evens” had something to do with this difficulty.

One of the older children offered a partial explanation: by turning over two heads, the number of heads-up coins decreases by 2; by turning over two tails, it increases by 2; by turning over one head and one tail, it stays the same. So the only possibilities after one move are to have 2, 4, or 6 heads face up, and all these are even.

I helped the children to generalise this to “an even change in the number of heads up” after any number of moves. Unexpectedly, one child objected to zero being considered an even number, which led me to take a detour. One serious concern I have with typical primary school mathematical education, which I share with Hung-Hsi Wu, is the absence of clear definitions. So I viewed this as an ideal opportunity to get the children to formulate the definition of an even number. I received these suggestions from the children:

  • a number that you can divide by two
  • a number that is a multiple of two
  • a number that ends in 0, 2, 4, 6 or 8

One child objected to the first definition, “because I can divide 1 by 2, to get 0.5”. So after some prompting, the children changed the first definition to “a number that you can divide by two and the answer is a whole number”.

We then worked through each definition in turn, testing it for whether zero was an even number. The oldest child provided a clear verbal proof that the first two definitions were equivalent, and a proof that any number meeting the third criterion also met the other two (but not vice versa).

The child who had objected to zero being even seemed to accept that by these definitions zero was even, but suggested that this would make \frac{0}{0} = 2, “which is not the case because \frac{0}{0} = 0“. I paused this question, writing on the board to return to, because we were in danger of getting too far off track.

For the last part of the session, I tried to get the children to reason algebraically, suggesting that – from their own definition – any even number can be written as 2n and, therefore because 4 + 2n = 2 \times 2 + 2n = 2(2+n), we still have an even number after any number of moves. I think I overestimated the preparedness of the children for this form of reasoning. The older kids were fine with this, one of the middle kids who had obviously seen but not understood algebraic notation before, upon seeing the n exclaimed “oh no, those numbers, I hate those numbers!” and the youngest had not met brackets in arithmetic expressions. Trying to cover all this at once was probably too much, but I hope we managed to get the basic ideas across.

Finally, I returned to the question, which I had rephrased, “what is zero divided by zero”. We looked at the equation 2 \times 0 = 0, and how – if the child who suggested it were correct, we could equally conclude from 3 \times 0 = 0  that the answer was “3” and from 4 \times 0 = 0 that the answer was 4. I suggested that there is no well defined answer, so we refer to it as undefined. Seemingly unsatisfied with this answer, one child left to get a calculator. I was hoping this to be my moment of triumph where the calculator reported “-E-“. Stupid calculator reported “0” (!) However, at least one child obviously could see what was going on, and asked whether 1 divided by zero would therefore be infinity. In the last few moments, I managed to explore this question with a subset of the children by looking at the growth of \frac{1}{1/n} as n increases in size, while the others packed away.

I’d view this as a mixed session. Lots of very interesting questions raised, not all answered satisfactorily, but at least most of the children were thinking mathematically.

Review: Love and Math by Edward Frenkel

I have a friend who thinks she does not understand maths (apologies to the American readers, I will use “maths” in this review rather than the “math” in the book title). She thinks maths is “all about numbers”. She thinks she’s not particularly good at it. She is wrong. Actually, I think she is remarkably gifted, primarily because she asks amazingly prescient questions. She has an inquisitive nature that I fear is often drilled out of most people by our “rigorous” schooling in the tedium of official school maths.

Over the past few months, I have spent quite a bit of time discussing maths with this friend, introducing her to set theory and trying to overcome a fear of algebra. (It is remarkable how many super bright people can reason about insanely complex general phenomena in their head without algebra in a way I never could, yet stumble over a simple equation!)

I saw this book reviewed elsewhere – I can’t quite remember where. It was a short but positive review, and the title is attractive to me, given my recent discussions. The author, Edward Frenkel, is a professor of mathematics at UC Berkeley, has made considerable contributions to mathematics himself, and also has a direct personal knowledge of many of the larger-than-life characters he talks about in the book. I knew nothing about his work, or his life, both of which turn out to be rather interesting, before reading this book.

The book is part autobiography and part introduction to the joy of mathematics. We learn about the author’s early career, the anti-Semitism he faced in Russia and his conversion as a teen from physics to mathematics, before the two come back together through quantum physics later in his career. We learn about his immersion in the Langlands Program.

As a book about “doing mathematics”, I think he does quite well at getting his feelings across. The detail of much of the mathematics is missing; sketches of subjects are given: Galois Groups, the Shimura-Taniyama-Weil Conjecture, Lie groups and Lie algebras, etc., but only very brief sketches. While this gives the reader insight into the depth of the subject areas covered by modern mathematics (and certainly helps to show the breadth of maths beyond school arithmetic) it doesn’t really show the reader mathematics. We read about proof, but we don’t experience proof. We read about revelation, but we don’t feel the revelation ourselves. We are told that mathematics rests on clear and unequivocal definitions, but these are absent from the book. Initially, this left me feeling unsatisfied. Slowly, I put aside my usual desire to understand everything I was reading, and began to enjoy the book as a novel, a love affair between the author and his subject, in which specific theorems appear only as partially sketched characters. From this perspective, the bulk of the book was very enjoyable, and provides insight not only into the beauty of mathematics, but also the ways in which mathematicians think. As an engineer, albeit one who has a lot of time for – and use for – pure mathematics, my way of thinking remains quite distinct from some of the methods and approaches outlined in this book. However, late in the book, where the author covers the links between this mathematics and quantum physics, I feel the unexplained mathematics again gets in the way. The author seems to acknowledge this, when he says (p.221), “All this stuff, as my dad put it, is quite heavy: we’ve got Hitchin moduli spaces, mirror symmetry, A-branes, B-branes, automorphic sheaves […] But my point is not for you to learn them all…”

The final chapter of the book takes a violent switch in direction as we learn of Frenkel’s artistic projects, in particular his film Rites of Love and Math. Had I known about this work beforehand, the book would have made a lot more sense! In this chapter, Frenkel provides his philosophical opinions on the nature of the material world, the world of our consciousness, and – in his view – the entirely separate Platonic world of mathematics. He says his approach to making the film was “let the viewers first feel rather than understand” – much the same could be said of this book.

The experience of reading this book has got me thinking in two directions.

Firstly, this is not the book I initially expected from the first few chapters. But how easy would it be to write a book which both conveys the joy of solving problems, say in Lie algebras, which Frenkel explores in some depth, and at the same time covers enough mathematics for a reasonably rigorous approach to the problem for a general reader? Perhaps such a task is impossible, and I ask too much. Does this mean that the beauty of modern mathematics is necessarily harder to access than than the beauty of modern art? Does this mean that we must necessarily pay the price of “boring maths” in order to be able to access “beautiful maths”? Or is there actually no “boring maths” and the prerequisite knowledge can also be described in a way that appeals and excites, even if not in one volume. I hope for the latter.

Secondly, the book has made me realise that I am far less certain of my own philosophical viewpoint on reality, Platonism, and knowledge than I used to be. It has made me want to explore these issues again and read more. Perhaps I will start with Penrose.

Is this the book I wanted for my friend? Probably not, but it’s certainly stimulated my thinking, and probably would also stimulate the thinking of others, regardless of their current level of mathematical maturity.

Math Circle Session 8: The Circle Game & Puzzles

Today was the 8th session of our math circle for primary school children. We only had four of the original six children today, for a variety of reasons, but it seemed to work well.

We first ran through a puzzle we set the kids to think about over the last week. The idea of the puzzle, taken from Martin Gardner’s book, is as follows. Take a piece of paper shown below. Imagine this is a map, that you have to fold up (I find folding up maps really hard!)

Screen Shot 2015-03-17 at 17.10.16You have to fold only along the lines shown, and must form a folded “map” with the 1 on top, facing up. The next number down after the 1 must be the 2, after the 2 must be the 3, and so on until the 8 lies on the bottom.

Some kids had tried this at home, some had not (as a policy, we never make “homework” compulsory) but none had found the solution, so we covered this today.

We then moved onto the main part of the session, “The Circle Game”, adapted from Rozhkovskaya’s book. The kids were each given a worksheet consisting of 16 points equally spaced around a circle, and asked to play a game in pairs. The rules of the game are as follows:

  • take it in turns
  • on each turn, connect one point to another point with a straight line; any pair of points can be chosen, but no line drawn can cross another line already drawn

The last player to be able to draw a line is the winner.

The kids enthusiastically played this game, trying to outsmart each other. However, at the end of play, we had three games in which the first player to move had always won (though the points selected were different). I also asked the kids to count the number of moves made, which was found to be 29 in each game. The question was posed: “why does the first player always win?”

One child quickly pointed out that if the number of moves is an odd number, it must always be the first player who wins. Another child suggested that we should try 17 points around the circle as then “it would be an even number of moves and Player 2 would win”. I suggested that instead of making the game more complicated, we simplify it to the smallest possible number of points on the circle, which the children identified as two points.

Children very quickly found that two points would always result in one move, three points in three moves, and four points in five moves. At this point the child who suggested that 17 points would result in a win for Player 2 withdrew his opinion, and all children were convinced that the number of moves would always be odd and therefore Player 1 would always win.

We then looked at generalising the number of moves required for n points. Children very quickly noticed that the number of moves increased by two for each new point added. Due to the significant difference in ages in the group, I wasn’t sure how comfortable children would be with an algebraic generalisation of the result, but it turns out that even the youngest was comfortable with the formula moves = 2 \times points - 3, which they got to via my suggestion that if the numbers were going up by two each time, try looking at the relationship between moves and 2 \times points.

I was surprised by how quickly we had reached this point in the session, so I rounded off the exploration by demonstrating an inductive proof of this formula in graphical form on the whiteboard (see, for example, Theorem 1.8 at

One child was keen to play this game with his dad, and it was suggested that the “youngest goes first” rule should be applied, possibly after agreeing a prize for winning!

With the remaining spare time for the session, we set the following puzzle – also from Gardner’s book – to begin during the session and to continue to think about over the week. Consider the grid below.

Screen Shot 2015-03-17 at 17.35.27The puzzle is to fill each empty cell with a single digit, so that the bottom row forms a number between 0 and 10^{10}-1. The challenge is that the digit under the heading zero must equal the total number of zero digits in this number, the digit under the 1 must equal the total number of 1 digits in this number, and so on. I thought it might be hard for the kids to understand the puzzle, but actually they understood it quite easily. They were amazed when I told them that the answer is unique: there is only one of the 10 billion possible numbers that works! It was interesting to observe the different reactions to this: some kids decided they’d never be able to find the solution, and therefore decided not to put much effort into the task. Others decided this was a challenge, and kept plugging away. One child was determined to take it home so that his dad would help.

Overall, I think this session went very well.

Update: I received an email tonight from the school headteacher with the solution to this number puzzle. I love that he’s so into it!

Math Circle Session 7: Combinatorics, Pascal’s Triangle, and the Sierpinski Triangle Again

Today saw Session 7 of our math circle. No blog on Session 6, as I was away for work – my wife explored The Towers of Hanoi in my absence.

Today’s session was based on a combination of two sessions in Rozhkovskaya’s book. The idea is as follows. Imagine a maze at a tourist attraction. The maze is such that there is one entrance, but seven exits labelled 0 to 6. Imagine that you are standing facing north at the entrance. At this entrance, and at each intersection, you can move ahead-left on a bearing of 315 degrees by one unit or ahead-right on a bearing of 45 degrees by one unit until you meet the next intersection or exit. Thus the maze decomposes into 7 levels (including the entrance and exits). Now further imagine that you must roll a die at each decision point, taking the left route if you roll even and the right route if you roll odd.

The first question: If I were an ice-cream vendor, where should I place my van to capture most people coming out? Children had an intuitive feeling that most people would come out in the middle. Only one child (the oldest) offered a reasoned explanation for this: “we expect as many odd throws as even throws, which means the most likely exit would be the middle one”.

We got the children to experiment by rolling routes through the maze, and collected a tally of how many trials ended up at each exit, confirming this suspicion. Children naturally noticed that exits zero and seven were unlikely, as they would require rolling “all odd” or “all even”.

This naturally led onto the next part of the exercise, to annotate each decision point with the number of ways in which it could be reached. Children found this hard, and made many mistakes, because they were trying to count all the ways to reach a given decision point from the maze entrance from scratch for each decision point. I explained that there are only at most two ways to reach any decision point, so it is sufficient to sum the possibilities found to these two preceding points.

Thus, Pascal’s triangle was revealed. We asked the kids to sum the rows of Pascal’s triangle, and they discovered they summed to successive powers of two.

Finally, we had the kids colour all the even numbers in a large version of Pascal’s triangle (pre-printed). One of the kids immediately recognised a fractal-like shape. Once complete, we pointed out the resemblance to Sierpinski’s triangle, created in Session 7. Oddly (to me) this wasn’t immediately apparent, even to the kids who were there.

Overall, I was a little disappointed with today’s session. I think the material was good, but it was a sunny day, and two of the kids showed no sign of wanting to be there, inside after school. This meant I had to spend a lot more time than I ever imagined trying to get these kids on task, meaning a lot less time to discuss the mathematics of what we were seeing, which was a shame. My one rule for Math Circle was that the kids should want to be there and, for whatever reason, I don’t think this was true for 1/3 of them this time. I hope it’s a temporary blip, but I need to ask for views from primary school teachers; I was again reminded today how different kids are to the adults I’m used to!