## MSLC Semina — Exploring Knights Tours

At the MSLC Seminar we had an “improvisational seminar” this week. We started off chit-chatting about various problems, and a theme emerged. One participant posed the following problem:

The game of Knight Placement is played on an 8×8 chessboard. Two players alternately take turns placing knights on the board. A move consists of adding a knight to the board, such that no knight is under attack. A player loses if they’re unable to place a knight. Who wins under optimal play?

I followed this question up with:

Suppose that the Queen of Chess has a garrison of twenty-five knights. The knights are kept on a 5×5 chessboard. One fine morning, the Queen shows up and orders the knights to all switch places, or be severely punished. Can every knight switch places simultaneously?

This got us thinking about knights tours. In a knight’s tour, a knight travels to every cell of a chessboard by visiting each square exactly once. Notice that if the 5×5 board has a knight’s tour, then the garrison can re-arrange themselves by each stepping along the tour.

We found a couple small boards with and without closed knight’s tours. Wikipedia turned our attention to this paper:

Allen J. Schwenk (1991). “*Which Rectangular Chessboards Have a Knight’s Tour?*” Mathematics Magazine: 325–332. (link)

Working through that paper might make a good session at MSLC Seminar. If anyone knows the history / providence of the puzzles above, I would be hear about them.

## Denlow Public School

I visited Denlow Public School and did two workshops for the Grade 4 and 5 students. The Grade 4s played with probability, learned to play Pig. This simple dice game has been subject to a lot of deep analysis. Some folks at Gettysburg College have given an optimal solution to the game.

The Grade 5 students learned about Cat’s Cradle. They were very excited, and wanted to learn more. Many students already knew a figure or two. We covered Half Second Star, Cup and Saucer, and Jacob’s Ladder. I’m told that they’re still playing with the string that I gave them.

## 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.

## Questions about Discs

Over the past couple weeks I’ve been asked a lot of questions about discs in Euclidean space. In this post we’ll be putting pennies on a table, refining covers of discs, and trying to cram lots of balls into high dimensional balls. Some open questions about putting pennies on tables occur below.

## The Pigeon Hole Principle

Below the cut are some pigeon hole related questions I collected together for a Math Circle at the Fields Institute.

## TwixT and Hex

Derek and I like to play abstract strategy games together. In highschool we both got hooked on Hex, a very elegant game with simple rules. Derek went so far as to beat Hexy on every single difficulty setting and board size — which is remarkable since Hexy plays well. Hexy has a really interesting approach to playing Hex that combines Shannon’s electrical resistance methods with automatic theorem proving techniques; you can read about it on the website. We usually play games on Richard’s Play by E-mail Server. So far we’ve played: Hex, Abalone, Y, Zèrtz, and TwixT. We both really like connection games ; Abalone and Zèrtz are the only non-connection oriented games on the list. We also used to try out games on PlayOK, back when it was called Kurnik and one could play Hex. We played a little Awari there, a beautiful african game I could never get the hang of.

A couple days ago, Derek started up play by e-mail boards that I haven’t got around to moving on yet. One 11×11 Hex board and a TwixT board. We’ve only played two games of Twixt thus far and I’m not sure that it suits my tastes. This third round might be the one that decides how I feel about the game. Right now I find the adjacency structure put on the grid by TwixT aesthetically unappealing. There are too many acute and obtuse angles for my liking.

Once we’ve played out the two matches, hopefully I’ll post a commentary.

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