Domino tiling.

What can you cover with dominoes?

Task

Supplies: a coin, dominoes, a chessboard, and masking tape. If you don't have dominoes, cut out rectangles that would cover two squares of the chessboard. 

Preparation: use the masking tape to select a smaller part of the chessboard. Start with putting tape around  3x3 square. 

What to do: 
- ask your child if she can cover the 3x3 square with dominoes. Let her explore (she will probably realize that every attempt would leave one small square not covered)
- place the coin on a small square then invite your child to cover the remaining 8 squares. When can she cover it? Now try placing the coin in a different spot. Is it possible to cover the remaining 8 squares?

How it went

I drew a 3×3 square on a piece of paper where the side of each smaller inside square matches the width of a domino instead of using a board.

Only Bel did this activity and she seemed equally interested in measuring and drawing the 3×3 square as in the activity itself. I shaded every other small square:

Can you tile this board with 2×1 dominoes?

Bel said there was always one square left. I then gave her a coin and ask her to put it on a random square, then figure out whether she can cover the remaining 8 squares with dominoes, she played with it a little bit. I showed her that she could keep track of her progress by drawing a smaller 3×3 square, and putting checkmarks or frowny face, depending on the success. She did figure out that, if the coin was on a dark square, it was possible to cover the rest with dominoes and if it was on a white square, she couldn’t do it. She did not have an explanation for it though.

We did a little bit of the 3×4 extension below, but Bel became bored very fast, perhaps because of the overwhelming number of ways to put two coins on the board.

Extensions

This can be extended as much as you want:

  • use the masking tape to get a 3×4 rectangle and place two coins on two small squares.
  • Can you cover the remaining 10 squares with dominoes? When can you do it, when can’t you? It will depend on where you put the coins – can you figure it out?
  • Do larger rectangles.

Next week, we will revisit and play with variations of previous activities.

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