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First, draw out a 10x10 grid. Take the shape below and see how many you can fit in the 10x10 grid. It should take up 6 grid squares. Also, it can be rotated. Hexomino

1) How many can you fit in the grid?

2) Can you prove that this is the most possible?

All credit goes to "Math Nite" at the University of Calgary

Rubio
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1 Answers1

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1)

I managed to put 16 shapes into the image Solution

2)

This is the maximum possible, as there are only four squares left

Halvard Hummel
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  • This was the same solution that I had. So far, this is the only one I've seen with 16 (without just rotating it.) –  Oct 19 '17 at 14:16
  • @Faraz there are only 4 grids squares left, so obsiously it is impossible to include another piece of 6 squares. Even if the shape was other, the maximum number of figures you can fit is 16, as 16*6=96, so you will never be able to include another figure of 6 squares, unless you have 102 blanks spaces instead of 100. – D. Mellow Oct 19 '17 at 14:25
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    I probably didn't say that very clearly. What I was trying to say is that I know 16 is the most, but this is the only configuration I've seen for 16 so far. @D.Mellow –  Oct 19 '17 at 14:38
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    Alright, now do the picture with only 4 colors on the map ;) – phroureo Oct 19 '17 at 16:32
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    @Faraz I verified using a computer program that the solution is essentially unique. It found only this configuration and its mirror image. – Jaap Scherphuis Oct 19 '17 at 22:12
  • I guess this is the only possible way to do it then, good job! –  Oct 20 '17 at 13:40
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    @phroureo There you go – Halvard Hummel Oct 20 '17 at 14:09
  • My program agrees the solution is unique. I searched a bit further, trying to find a larger square with a higher packing density than the current 10x10 with 96% density. No joy so far, busy searching squares 24 through 29 with six copies running at once. Can someone come up with a proof that 96% is maximal to free up some of my CPU cores please... – theonetruepath Dec 09 '17 at 09:56
  • This hexomino, like all hexominoes, can tile the plane. However, it cannot tile a rectangle. It follows that the tiling density for an arbitrarily large square can be arbitrarily close to, but not equal to, 100%. – nickgard Dec 09 '17 at 18:03
  • You're right of course... that means my search should continue. But finding "the smallest square with tiling density higher than 96%" may well be out of range of a computer search. In fact someone with a knack at hand solving might find the answer more easily/quickly. – theonetruepath Dec 09 '17 at 18:18