## Q8(b) from 2014 Final Winter

Exercise 8bAssume is an matrix with and has the property that each entry in the first row equals the sum of the entries in the column below it. Prove that .

## Two nice facts about games.

Since giving my talk last week about games, I’ve been on a bit of a mathematical gaming kick, and would like to share some of the gems I’ve come across. We’ll give a beautiful argument due to Gale about his game Chomp, and an exceedingly clever argument due to Hochberg, McDiarmid, and Saks which applies Sperner’s lemma to the Game of Y.

## Mathematics and Games (Kangaroo Math 2014)

On Sunday, March 23rd, 2014, I gave a talk to parents of kids writing the 2014 Kangaroo Math contest. The slides are available here: Kangaroo 2014 Slides. The content of the talk was a discussion of Tic-Tac-Toe in disguise, and a proof of the winning strategy for Nim.

Click more for further information.

## Dodecahedral Crafts

This evening I made some dodecahredral crafts to show my mentoring students. This week we’re proving that the dodecahderon actually exists, by constructing it explicitly. The model on the right illustrates our proof splendidly. For instructions, check out Laszlo Bardos’ site CutOutFoldUp.com. The dodecahedral calendar is from Marlies’ Crafts.

## 2014 Mentoring

I’ve added a page about the mentoring project that I’m working on with three Gr. 12 students this semester. I’ll be updating the page as we go. For more information see Mentoring — 2014. From the introductory remarks:

The plan for the semester is an ambitious one. We’re going to understand the structure of all the regular convex polytopes in all dimensions, and build up a intuition for dimensions greater than three. We’ll spend most of our time learning the tools we need to understand how a geometric object can be pieced together. These tools will include vectors, metric spaces, symmetry groups, and simplicial complexes. I’m taking a very broad view of what constitutes a tool and counts as information about a space. The final result of the project will be a poster presentation about the solids and some 3-dimensional “nets” of the 4-dimemsional solids (the simplex, cube, cross-polytope, and 24-cell).

I can’t resist saying things about hyperbolic geometry. Therefore, if we get to the end of the proposed project, we’ll take a stab at using out high brow high dimensional intuition to understand the Gieseking manifold. There is more than enough stuff to say about the platonic solids, so we’ll see how far we get.

## Singularities I.1

These are some notes that I’m writing up on Gromov’s papers on singularities of maps. This first post will look at some of the introductory material in: Singularities, Expanders and Topology of Maps. Pt 1. These notes will be a partial introduction to what is going on.

## Of Waists and Spheres

These are some notes that I’m writing up on Gromov‘s *waist inequality*. We’ll look at some standard material about the Borsuk-Ulam theorem and finish with a nice application of the inequality.

## Dwarves with hats

This afternoon Mike Pawliuk told me the following problem:

A thousand dwarves are wearing standing around wearing red or blue hats. No one can see their own hat and no one can communicate with anyone. Can the dwarves devise a strategy to form a line so that all the red hatted dwarves are on one end, and all the blue hatted dwarves are on the other?

While thinking over the solution, it occurred to me that one can obfuscate the problem as follows:

Consider a group of n dwarves as above. Can the dwarves devise a strategy such that all but a uniformly bounded number of dwarves know the colour of their own hat correctly with absolute certainty?

I can’t see a route to the solution of the second problem that doesn’t go through Mike’s puzzle. This is a terrible case of solution induced blindness. Perhaps someone can point out a more direct solution.

**Edit**: After some reflection, Mike and I have concluded that this problem is too easy without placing restriction on how the dwarves can move around, since a lot can be encoded by in their movement patterns. It’s very easy to get absolute certainty for all but one dwarf as posed.

For the record, here is another dwarf problem that Michelle BouĂ© told me about when I was an undergrad:

Suppose that three dwarves with red or blue hats are sitting in a circle. They decide on a strategy to guess the colour of their own hat without communicating. They all guess simultaneously. Can they do better than guessing uniformly at random?

## Of Loewner and Besicovitch

I’d like to share some of the notes that I’m writing up about systoles. After a little bit of preliminaries we’ll see a slick proof the systolic inequality in the torus case.

The systole of manifold is the length of the shortest non-contractible curve in the manifold. Systoles hard to estimate. In general there are many many non-contractible curves, and its not easy to track down which one should be smallest. If someone hands you a donut, you’ll visually guess the systole correctly. If someone hands you a coffee cup, it’s still clear. Once you get a generic metric, you’re in deep water. Loewner‘s theorem gives us an upper bound on the systole a Riemannian 2-torus (generalized donut / coffee cup case).

## The Rotationally Distinct Ways to Label a Die

I’m giving a talk at the Canadian Math Camp this year. I’ll be showing the kids of how to count the number of ways to label a six sided die up to the rotational symmetries of the cube. Here is the handout for the talk with questions about dice labellings, the 15-puzzle, and permutation groups.

For the curious the labellings are below the cut. Please note that there are typos in the table below. Alex Fink kindly pointed them out and they will be fixed eventually. For now they are an exercise in keen observation.

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