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We have a $3 \times 3$ square grid, and we must color $5$ squares in this grid, but each colored square must be connected to all other colored squares (There must be one connected shape, not multiple shapes). How many different ways can we color this grid? Thanks in advance.

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OK, so the shapes are all 'pentominoes':

https://en.wikipedia.org/wiki/Pentomino

The pentominoes that fit into a 3x3 square are:

F,P,U,T,V,W,X,Z

Assuming rotation and mirroring can all amount to different colorings, we thus have the following possibilities:

F: 8 possibilities (mirror is different, and each of the four rotations is different)

P: 16 possibilities (mirror is different, and each of the four rotations is different, and it can be put in two different pairs of columns/rows)

U: 8 possibilities (each of the four rotations is different, and it can be put in two different pairs of columns/rows)

T: 4 possibilities (just 4 rotations)

V: 4 possibilities (just 4 rotations)

W: 4 possibilities (just 4 rotations)

X: 1 possibility

Z: 4 possibilities (2 rotations + mirror)

For a total of: 49 possibilities

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