Apologies to any Antipodean Swifties arriving on this page. Today’s puzzle is about tiles, and whether or not you can solve it swiftly.
The puzzle concerns black and white tiles on a 4×4 grid. 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 find a way to tile the grid such that:
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?
I’ll be back with a solutions at 5pm UK.
NO SPOILERS Please discuss your favourite tilers, Tylers, Taylors and/or swifts instead.
Today’s puzzle was devised by maths outreach legends Katie Steckles and Peter Rowlett, who together with Sam Hartburn and Alison Kiddle are the authors of Short Cuts: Maths, which provides bite-sized introductions to many mathematical ideas. One topic included is matrices and block designs, an introduction to which is this very puzzle.
Katie and Peter are also both 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.
