Earlier today I set you this puzzle, about the tiling of a 4×4 grid. It requires a swift 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 colour, or it has a different colour.
For example, in the checkerboard pattern, below left, each tile in the top row has a tile in a different colour below it. Likewise for row 2 and row 3.
For the grid on the right, two of the top row tiles have a different colour directly below them, and two have the same colour directly below them. For the second row, again, two have a different colour below them, and two the same colour. The pattern breaks down, however, in the third row, where all four tiles have a different colour below them.
Project tile
Your task is to you find a way to tile the grid such that both of these conditions apply:
1) For every row (except the bottom one), two tiles have the same colour directly below them and two tiles have a different colour.
2) For every pair of adjacent columns, (shown below) two tiles in the left column have the same colour directly to the right and two tiles in the left column have a different colour to the right.
If you found that easy, here’s one for the pros: can you tile an 8×8 gird the same way? That is, such that for each pair of adjacent rows/columns matches, the tiles match in half the positions and differ in half of the positions?
Solution
Here’s a solution to the 4×4. Below shows all adjacent rows and columns and how you get two positions matching 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 in the bottom right put an inverted version (i.e. with black and white flipped). Neat!
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 livestreams and discussion, as well as a feed of interesting maths content from all over the internet. Visit patreon.com/finitegroup to sign up.
Katie and Peter, together with Sam Hartburn and Alison Kiddle, are the authors of Short Cuts: Maths, which provides bite-sized introductions to many mathematical ideas.
I’ll be back in two weeks.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.