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Programming Enigma Puzzles

6 April 2020

Posted by on **From New Scientist #1696, 23rd December 1989** [link]

Arranging and displaying the Christmas cards is always a problem. This year all our cards are either 10cm × 20cm or 20cm × 10cm. We managed to arrange them together like a jigsaw, just covering (without overlapping) a square piece of paper.

Then we found that there was no convenient place to display the square so we decided to cut it either horizontally or vertically into two rectangles. But no matter how we tried it was impossible to do this without cutting through at least one of the cards. So we cut the square into two rectangles in such a way that we had to cut through the minimum number of cards, but it still meant that we cut over five per cent of our cards.

How many cards did we receive this year, and how many did we cut?

**Enigma 192** was also called “Merry Christmas”.

[enigma544a] [enigma544]

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(See also:

Enigma 1653).The setter has managed to find a “fault-free” tiling of his square.

We know (see my notes on

Enigma 1653) that the following conditions must hold for ann × nsquare:So

nis an even number greater than 6. (i.e.n = 8, 10, 12, …).For an 8×8 grid we would need to cut (at least) 2 tiles. For a 10×10 grid, (at least) 3 tiles, etc.

Can we create a fault free tiling for an 8×8 grid, that requires us to cut (at least) 2 tiles?

Can we create a fault free tiling for an 8×8 grid, that requires us to cut (at least) 3 tiles?

And is it possible to create a fault free tiling for 10×10 grid that requires us to cut (at least) 3 tiles?

So this is impossible, as are any larger grids. The only possible solution is with an 8×8 grid, and a minimum of 2 tiles cut.

Solution:32 cards were received. 2 of the cards were cut.This numbers argument tells us that

ifthere is a solution to the puzzle, it must be using an 8×8 grid, and the minimum number of tiles cut by a fault line is 2.But is it possible to construct such a grid?

The following Python 3 program runs in 107ms.

Run:[ @repl.it ]Here is an example 8×8 grid where each fault line is bridged by at least 2 tiles.