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How many squares does a (regular) chessboard have?

I know its 204 but I have no idea how to justify it.

 

Also, how many rectangles does a (regular) chessboard have?

 Sep 13, 2020
 #1
avatar+555 
+2

See this for squares: http://puzzles.nigelcoldwell.co.uk/twentyseven.htm

 

For rectangles: 

 

We can choose the two verticals in \(\binom{8+1}{2}=36\) ways, and by symmetry, it is the same for horizontals. So 36^2=1296.

 Sep 13, 2020
 #2
avatar+205 
0

Thank you

TacoBell  Sep 13, 2020
 #3
avatar+555 
+2

No problem. The technique with choose binomials is very helpful in geometric counting/probability.

thedudemanguyperson  Sep 13, 2020

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