Maximum number of 5 cell figures on an 8x8 board

[jegues]

I wasn't entirely sure where to post this so I guessed here would be the best place.

See the figure attached for the problem statement.

How do you go about solving a problem like this? There must some sort of proof to go along with it explaining how this undoubtedly the maximum number of 5 cell figures, right?

I'm not doing a math degree or anything so maybe this is a common type of problem in a certain area of mathematics.

Can somebody help clear things up? A start at the problem would be nice!

[raa91]

This a probability problem. you have 16 cells out of which u need to take out 5 cells that are somehow connected. So find the total number of cases then eliminate those you don't need such as the corner cells. or you can simply count the cases...

[Unknown008]

Maybe these will help you

Polyomino - Wikipedia, the free encyclopedia

Pentomino - Wikipedia, the free encyclopedia

Those are called polyominoes. You are certainly familiar with dominoes (d - two, omino), well, the ones in your problem involves pentominoes (pent - five, omino)

[Wilmer]

If you want to try something else (kinda similar), go here:
Problem #19

[Opalg]

You can fit eight of these X pentominoes onto an 8x8 board, as in the attachment. I don't see any way to improve on that, but it would be nice to have a proof that this is the maximum number.

[Archie Meade]

I wasn't entirely sure where to post this so I guessed here would be the best place.

See the figure attached for the problem statement.

How do you go about solving a problem like this? There must some sort of proof to go along with it explaining how this undoubtedly the maximum number of 5 cell figures, right?

I'm not doing a math degree or anything so maybe this is a common type of problem in a certain area of mathematics.

Can somebody help clear things up? A start at the problem would be nice!

Mathematically speaking,
on an 8x8 chessboard, there are 5(12)+4 small square cells.

Therefore, at least 4 cells will necessarily be unoccupied.
Hence the maximum number of arbitrarily-shaped non-overlapping 5-cell figures is 12,
which Opalg has shown.

[undefined]

Alternate way to place 8 figures, of course the figures can be shifted to get some more arrangements.

Edit: sorry the image is so big, i don't have photoshop on this computer and it's a bit of a hassle to resize in a way that looks nice.