# Hexominoes Challenge: 15x15 - 15

The pentominoes are often presented in an 8x8 frame with an O-tetromino or something to make up the final 4 unit squares needed to completely fill the 64 square area. And there are a lot of pentomino puzzles based around the 8x8 shape, with the four holes in various different positions.

Fig. 1: Some examples with pentominoes. That last one is possible but you'll have to find the solution for yourself.

While playing around with the hexominoes, the closest I've found to an analogue of this is the 15x15 square with fifteen holes. As fifteen isn't divisible by 4 or of the form (4n+1) this prevents the configuration of holes from retaining the symmetries of the outer square, but there's still fun to be had. Provided, that is, that solving things with hexominoes is your idea of fun. I know it's not everyone's cup of tea.

The most obvious configuration is to have the fifteen 'missing' squares collected together in the centre in a 3x5 rectangle, as in the first image on this page. As far as hexominoes solutions go, this one's fairly forgiving owing to its straight edges externally and internally, and the lack of narrow bottlenecks. It serves as a good introduction for those just beginning with solving larger polyomino sets by hand. Then from there, several variations on that theme can be solved, with varying levels of difficulty.

If you're not too picky about preserving the two mirror symmetry axes then the possibilities are pretty much endless. Here's a few possibilities with the holes marked in grey, with difficulties ranging from 'tough' to 'tougher' to 'really tough'. Clicking on the image will show a solution to each. I think with there being 35 hexominoes, enumerating solutions to any one shape is far beyond the reach of your average computer.

(Click for solutions.)

And a final word of warning: Parity issues abound with hexominoes, and these kind of solutions are no exception. Some day I'll write up the explanation of parity imbalance that I needed and couldn't find when I first got into polyominoes in a big way, but today is not that day.

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Lewis Patterson. Last updated 22/02/22.