**From New Scientist #2241, 3rd June 2000** [link]

Place your finger in the starting box in the grid and follow each instruction. You will find that you visit each square before finishing in the appropriate box.

Now cut up the board into six rectangles each consisting of two adjacent squares. Then put them back together to form a new three-by-four grid with the starting square one place lower than it was before. Do all this in such a way that, once again, if you follow the instructions you visit each square.

In your new grid, what instruction is in the top left-hand corner, and what is the instruction in the bottom right-hand corner?

[enigma1085]

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I tackled this puzzle in two parts. The first part is to arrange the squares in the grid so that you can start in row 2, column 2 and complete the gird. The second part is to look at the grid made from the first part and see if it can be made from the original grid by cutting it into dominoes.

This Python 3 code runs in 124ms.

Run:[ @repl.it ]Solution:In the new grid the instruction in the top left-hand corner is “GO RIGHT TWO”, and the instruction in the bottom right-hand corner is “GO LEFT TWO”.There is only one way to make up the grid so that you can start in row 2, column 2 and complete the grid, even if you cut the original grid into single squares, so we know what the solution is, if it is possible.

There are two tiles that read “GO LEFT TWO”, so these tiles can be interchanged without changing the pattern. But by cutting the original grid into 2×1 “dominoes” only one of these tiles can appear in the bottom right-hand square, but there are multiple ways to make the cuts. We can cut the square into 2 2×1 rectangles and 2 2×2 rectangles which re-arrange to make the solution grid, as shown below:

The 2×2 tiles can be cut either horizontally or vertically to make dominoes, which gives us 4 different sets of dominoes that can be arranged to make the grid.