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

cosign Jan 31, 2022

#1**+3 **

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

BuilderBoi Jan 31, 2022

#2

#3**+2 **

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

#5**+2 **

_____ _____ _____ _____ _____

_____ _____ _____ _____ _____

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.

BuilderBoi Jan 31, 2022