Counting Squares

Probability Level 2

How many squares are there in the above image?

Hint: Think about the larger square.

The answer is 38.

1 solution

Kexin Zheng
Jun 9, 2016

If you consider the full 5 by 5 grid, it has $1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55$ grid squares.

We only need to count the number of squares missing due to the four corners and subtract it from 55 to get the answer. $1\times1: 4\ squares.$ $2\times2: 4\ squares.$ $3\times3: 4\ squares.$ $4\times4: 4\ squares.$ $5\times5: 1\ square.$ The total number of squares missing is $4+4+4+4+1 = 17.$ Subtract this from the total number of squares on a 5x5 grid, we get $55 - 17 = \fbox{38}$

Heehee, very nice! That's exactly what I was thinking of. Trying to do it otherwise can be quite tedious.

Do you think the hint gives too much away?

Chung Kevin - 5 years ago

Nice question. My sense is that the question is better without the hint.

Brian Charlesworth - 5 years ago

Do you suggest to remove it completely?

Okay, I changed it. How is it now?

Chung Kevin - 5 years ago

@Chung Kevin – I would. Realizing that the 'subtraction' method is the best approach is what makes this question worthwhile, and for those who solve it the tedious way, they will learn something new from Kexin's solution.

Edit: Well, I would have removed the hint completely, but it depends on what level you want the question to settle at. With the hint it will probably be level 1 or 2, and without it level 3.

Brian Charlesworth - 5 years ago

Honestly, I only realized that the subtraction method works because of your hint :P

Kexin Zheng - 5 years ago

Counting Squares

The answer is 38.

1 solution

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