10
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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

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19
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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

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  • $\begingroup$ This was the same solution that I had. So far, this is the only one I've seen with 16 (without just rotating it.) $\endgroup$ – user41716 Oct 19 '17 at 14:16
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    $\begingroup$ 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 $\endgroup$ – user41716 Oct 19 '17 at 14:38
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    $\begingroup$ Alright, now do the picture with only 4 colors on the map ;) $\endgroup$ – phroureo Oct 19 '17 at 16:32
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    $\begingroup$ @Faraz I verified using a computer program that the solution is essentially unique. It found only this configuration and its mirror image. $\endgroup$ – Jaap Scherphuis Oct 19 '17 at 22:12
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    $\begingroup$ @phroureo There you go $\endgroup$ – Halvard Hummel Oct 20 '17 at 14:09

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