May 28, 2024


Earlier today I set you this puzzle, about tiling a 4×4 grid. This calls for a quick preamble, so here we go again.

Consider the image below, which highlights adjacent rows in the grid.

For each cell in a top row, there are two choices for the cell directly below it: either it has the same color, or it has a different color.

For example, in the checkerboard pattern, lower left, each tile in the top row has a tile of a different color below it. Similarly for row 2 and row 3.

For the grid on the right, two of the top row of tiles have a different color directly below them, and two have the same color directly below them. For the second row, again, two have a different color under them, and two the same color. However, the pattern breaks down in the third row, where all four tiles have a different color underneath them.

Project tile

Your task is to find a way to tile the grid such that both of these conditions hold:

1) For each row (except the bottom one), two tiles have the same color directly below them and two tiles have a else color.

2) For each pair of adjacent columns, (shown below) two tiles in the left column have the same suit directly to the right and two tiles in the left column have a else color to the right.

If you found it easy, here’s one for the pros: can you tile an 8×8 belt the same way? That is, such that for every pair of adjacent rows/columns that match, the tiles match in half the positions and differ in half the positions?

Solution

Here is a solution for the 4×4. Below shows all adjacent rows and columns and how you get two positions that match and two that don’t.

To get an 8×8 that follows the same rules, place three of these 4x4s in the top left, bottom left, and top right of an 8×8 grid. And on the bottom right sits an inverted version (ie with black and white reversed). Tidy!

Thanks to Katie Steckles and Peter Rowlett for today’s puzzles. They are part of Finite Group: an online community for people interested in playing with mathematical ideas – with monthly live streams and discussions, as well as a stream of interesting math content from all over the internet. Visit patreon.com/finitegroup to report.

Katie and Peter, along with Sam Hartburn and Alison Kiddle, are the authors of Shortcuts: Mathematics, which provide bite-sized introductions to many mathematical ideas.

I’ll be back in two weeks.

I’ve been posting a puzzle here on alternate Mondays since 2015. I’m always on the lookout for great puzzles. If you want to suggest one, email me.



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