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# Is there a quicker way to count this?

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Four squares are joined together to form an L-shaped piece. In how many ways can you place the L-shaped piece on a  5 x 5 grid? The L-shaped piece can be rotated and/or reflected.

I started counting normally just fitting it wherever but I don't think this is efficient. So I started finding all the cases that won't work but this is taking to long. Can anybody help me?

Is there a quicker way to count this?

-cosign

Jan 31, 2022

#1
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The 4 squares are unit squares that form an L shape that is 3 high and 2 wide right?

Jan 31, 2022
#2
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Ye like this

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the dots mean 1 block

Jan 31, 2022
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There are \(4\) ways to arrange the L so the bottom \(2\) peices are in the bottom row.

There are \(3\) rows that you can do this on, so there are \(12\) ways to arrange the L this way.

There are \(3\) additional ways to rotate them \(90^\circ\) with the same concept. This means that there are a total of \(48\) ways with rotations.

There are \(3\) ways to reflect the L , with each one having \(12\) ways to arrange them.

This makes for a total of \(\color{brown}\boxed {84}\) ways.

BuilderBoi  Jan 31, 2022
edited by BuilderBoi  Jan 31, 2022
#4
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Woah.... that is really smart!

I have not thought about it that but what do you means the bottom 2 pieces are in the bottom row

cosign  Jan 31, 2022
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_     _     _     _     _

_     _     _     _     _

X

_     _     _     _     _

X

_     _     _     _     _

X    X

_     _     _     _     _

Each X represents where one block would be. The other possible combinations would involve transforming the L \(1-3\) units to the right.

Jan 31, 2022
edited by BuilderBoi  Jan 31, 2022
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oh ok!

cosign  Feb 1, 2022