Earlier today we had this puzzle about tiling a 4×4 grid. I need a quick introduction, so I’ll explain it again.
Consider the image below that highlights adjacent rows in a grid.
For each cell in the top row, you have two choices for the cells directly below it: the same color or a different color.
For example, in the checkerboard pattern on the bottom left, there is a different colored tile below each tile in the top row. The same goes for the second and third lines.
In the right grid, two tiles in the top row have different colors directly below them, and two tiles have the same color directly below them. Even in the second row, the two below are different colors and the two are the same color. However, on the third row, the pattern breaks and all four tiles below it are a different color.
project tiles
Your task is to find a way to tile the grid such that both of the following conditions apply:
1) For all rows (except the bottom row), two tiles have same The two tiles have a color just below them. different color.
2) For each pair of adjacent columns, the two tiles in the left column (shown below) have same Immediately to the right are the colors, and the two tiles in the left column have different Color on the right.
If you think it’s easy, here’s how to do it professionally. Can I arrange an 8×8 grid in the same way? That is, if each pair of adjacent rows/columns matches, will the tiles match in half of the positions and differ in half of the positions?
solution
Here is the 4×4 solution. Here’s how to get all adjacent rows and columns and the two matching and two non-matching positions.
To get an 8×8 that follows the same rules, place three of these 4x4s at the top left, bottom left, and top right of the 8×8 grid. And in the bottom right, place the inverted version (i.e., the white and black are reversed). Neat!
Thanks to Katie Steckles and Peter Rowlett for creating today’s puzzle. They are part of the Finite Group. Finite Group is an online community for people interested in playing with mathematical ideas. We also offer monthly live streams and discussions, as well as a feed of interesting math content from across the internet.visit patreon.com/finitegroup Sign up.
Katie and Peter, along with Sam Hartburn and Alison Kiddle. Shortcut: Mathematics, It introduces many mathematical ideas in an easy-to-understand manner.
I’ll be back in two weeks.
Since 2015, I have been posting puzzles here every other Monday. Always looking for great puzzles. If you would like to make a suggestion, please email me.
