# Math on a Chess Board

M. C. Escher

Sometimes, when we have friends over, Rocket Boy gets out his chess board and his favorite chess set for a friendly game. But the kids don’t always end up playing chess. Instead, they build other, pretty elaborate, games on and around the board.

For example, once they played “tea-time in the chess land”. Another time the kids were building something with LEGOs on the chess board. One little child built “families” out of chess figures, sorting and combining them this way and that.

A chess board, as it turns out, can be used for more than playing chess (or checkers). It can be used for mathematical explorations.

Martin Deutsch. The Chess Game.

One of the first that come to mind is the classic story that involves a (not very math-smart) king, a clever (and very math-savvy) servant, a chess board and grains of wheat or rice.

This one story can be replayed and re-imagined many different ways. Some time ago I asked Rocket Boy what would he preferred, if he were to win a “major prize” – 1 million LEGO blocks all at once or 1 block the first day, 2 blocks the second day, 4 blocks the third day, etc? We also applied this hypothetical scenario to cookies, allowance money, and Pokemon cards. We’d get the board out and start counting off the objects and putting them in the squares.

What if you create your own rule for filling up the squares? For example, 1 LEGO block in the first square, 1 – in the second, 2 – in the third, 3 – in the fourth, 5 – in the fifth, 8 – in the sixth… How many blocks will be in the 8th square? Do you recognize the pattern?

What if you start with the first square in the first row and put 1 red block in it. In the second square of that same row put 2 red blocks. 3 red blocks go in the third and so on until 8 red blocks go into the 8th square. Now, let’s fill up the second row. The first square would have 1 red block and 1 blue block. The second square – 2 red blocks and 2 blue blocks. The third square – 3 red blocks and 3 blue ones… Once you fill up the entire second row (8th square would have 8 red blocks and 8 blue ones), move on to the first square of the third row and stack 1 red block, 1 blue block and 1 green block. Continue the pattern. Do you recognize it? How many blocks will be in the 8th square of the 8th row?

What other patterns can you and your child create? Can you guess the rules for these patterns?

Another classic chessboard math puzzle is chessboard dominoes. Can a chessboard be completely covered with domino tiles (1 square wide by 2 squares long) so that there are no gaps and the tiles don’t overlap? Easy-peasy! But what if you remove two diagonally opposite corners (Hint: you can place guarding knights on those corners to serve as markers instead of breaking out your scroll saw). Try several different approaches. What do you notice?

Somehow the dominoes puzzle reminded me of a game I used to play as a child with my brother. The game, quite popular in Russia, is called Ugolki (literally: corners). Here’s a quick explanation in English. For younger children this game can be played with fewer number of pieces, say 6 per side.

While searching for more ideas for math games and puzzles on chessboard that would be suitable for younger children, I came across an app called Tangram Chess (\$0.99). Similar to Ugolki, a player has to move her pieces across the board and re-assemble them into the same formation. Except these pieces are tangram tiles. So they can be moved using translation, rotation and reflection. The game can be played in a single-player or multi-player modes. I’m definitely going to play this game with Rocket Boy.

Another game I loved playing as a child was The Knight’s Tour. In it, a knight is placed on a square of an empty board. It has to move according to how a chess knight moves and visit every square on a board only once. Can it be done? An 8×8 board can feel a bit overwhelming, so how about starting with a smaller board? Or for more of a challenge, how about a 10×10 board? What about a board that is rectangular, instead of square?  This game can be played online, but also on paper or, using reusable stickers, on a regular chessboard.

And how about playing the Free the Clones game? In many ways it’s a much easier game for young children to handle. Besides, it seems sooooo very simple!

A friend recently told me that her child built a chessboard and the chess pieces in Minecraft and uses it to play games. This sounded very cool! And here’s something else you can do, especially if your child is into Minecraft, but you are concerned with the screen time she’s getting. You can print Minecraft characters or blocks (we use MCPapercraft app), then cut them out and fold and glue together to make your own Minecraft-themed game pieces and learn about geometric solids.

Now that you have your own chess pieces, maybe you’d like to create your own chessboard. Let’s pick two colors (for now). Now, let’s go crazy and color the board the way we want. Maybe one row will be all one color and the other – all different color? Or maybe the colors would alternate every two squares? Or we’d color diagonals all the same color. Can you figure out how many ways there are two color an 8×8 board using two colors? Perhaps, to get started, we’ll try a smaller board. How many ways can you color a 2×2 board with two colors? What about a 3×3 board? What do you notice? (This idea is inspired by Richard Evan Schwartz’s book Really Big Numbers)