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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
avatar+2455 
0

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

 Jan 31, 2022
 #2
avatar+91 
+2

Ye like this

 

.

.

.    .

 

the dots mean 1 block

 Jan 31, 2022
 #3
avatar+2455 
0

 

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
avatar+91 
+2

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
 #5
avatar+2455 
0

_     _     _     _     _

 

_     _     _     _     _

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
 #6
avatar+91 
+1

oh ok!

cosign  Feb 1, 2022

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