# How Many Squares (with Answers)

This seems to be an odd form of Internet troll-bait: puzzles for which lots of people will claim to have a better answer, but if asked will not tell.

find all 30 squares

A "square" means any set of edges in the diagram which together form the 4 edges of a square (when all other edges are ignored).

We can answer the puzzle by starting with a simpler version and enlarging it one step at a time:

N=1:

There is 1 square.

N=2:

1 big 2×2 square plus 4 little squares = 5.

N=3:

find all 14 squares

1 big square, 4 ways to find a 2×2 square, and 9 little squares. 1+4+9 = 14.

Here we can see a pattern: 1+4+9 is 1^{2}+2^{2}+3^{2}. So the number
of squares is a sum of squares. Cute.

N=4:

find all 30 squares

1^{2}+2^{2}+3^{2}+4^{2} = 1+4+9+16 = 30. This answers the first question,
the diagram at the top of this page.

Other more complex versions of the same puzzle are often seen.

find all 40 squares

(or, if troll-baiting: How many squares can you find in
this diagram?)

This seems to be the most popular at the time I am writing this. The
answer is 40: the 1^{2}+2^{2}+3^{2}+4^{2} = 30 of the original puzzle,
plus 5 for each of the smaller 2×2 puzzles that have been added.

find all 42 squares

This is a tiny bit different from the popular 40-squares version. Again we have 30+10, but there are 2 more:

for a total of 42.

find all 67 squares

Here we find the original 30 plus the 29 squares in this "cross" puzzle:

find all 29 squares

which you could count as five copies of the 2×2 puzzle plus 4 more of the larger size square: 5×5 + 4; or alternatively you could count it as 20 little squares + 5 big squares down the central column and 5 big squares across the center row, minus 1 to avoid counting the center twice, either way that gives 29.

So we have 30 + 29, but when we overlay the "cross" on the 4×4 grid we get 4 more little squares:

plus 4 more squares of this size:

for a grand total of 67.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2018 Aug 27. s.11