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9

Is it possible? Why?

8

(I don't think I've seen this before, but having figured it out and written it down I have a bit of a sense of déjà vu...) Suppose we A bit of notation: Now So Therefore In other words, Here's a fairly crappy diagram that may help follow the above:

6

Proof without words:

6

The easiest way to count is

4

By the look of it, this puzzle comprises one of each of the 12 different pentomino shapes. In which case, an example packing can be found on the Wikipedia page for 'Pentomino', as follows: If you would like a little history, the 6x10 case was first solved as far back as 1960 by Colin Brian Haselgrove and Jenifer Haselgrove, and there are apparently 2,339 ...

4

This puzzle of mine appears to have languished with no solution for some time. I don’t remember the solution I initially had in mind, but I thought of one recently that might be of interest, so answering my own puzzle here. It suffices to show that for any $n$, there exists an $n$-gon inscribed in the unit circle with pairwise-distinct rational distances, ...

1

What we find from this recursive process is Now In conclusion

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By visual inspection of the diagram below it can be seen that a 3x3 square is divided into 7 rectangles whose sides are in 2:1 ratio.

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I wrote a solver for this (and related polycubes) using or-tools. You can see it here https://github.com/MrBenGriffin/or-tools-fun There are many related puzzles! A standard pentomino puzzle is to tile a rectangular box with the pentominoes, i.e. cover it without overlap and without gaps. Each of the 12 pentominoes has an area of 5 unit squares, so the box ...

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