How many squares do you see?

How many squares do you see?

  • In a square of 2 × 2 2\times 2 there are 1 + 4 1 + 4 squares.

  • In a square of 3 × 3 3\times 3 there are 1 + 4 + 9 1+4+9 squares.

  • In a 4 × 4 4\times 4 square there are 1 + 4 + 9 + 16 1 + 4 + 9 +16 squares.

  • In a 5 × 5 5\times 5 square there are 1 + 4 + 9 + 16 + 25 1 + 4 + 9 + 16 + 25 squares.

  • In a square of 10 × 10 10\times 10 , how many squares are there?


The answer is 385.

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3 solutions

Matin Naseri
Jan 31, 2018

solution 1 \text{solution}^{1}

1+4+9+16+25+36+49+64+81+100=385 \text{1+4+9+16+25+36+49+64+81+100=385} .

solution 2 \text{solution}^{2} .

i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 \quad \sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}

1 2 + 2 2 + 3 2 + + 1 0 2 = 10 ( 11 ) ( 21 ) 6 = 385 1^2 + 2^2 + 3^2 + \ldots + 10^2 = \dfrac{10(11)(21)}{6} = \boxed{385}

Nice solution

Harison Allan - 3 years, 4 months ago

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You are welcome.

Matin Naseri - 3 years, 4 months ago

[ A w e s o m e ] ( s o l u t i o n s ) \color{#3D99F6}{[Awesome](solutions)} .

But one solution is enough!

Sarfarz Saifie - 3 years, 4 months ago

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You are welcome.

Yeh but I add two solution.

Matin Naseri - 3 years, 4 months ago
Munem Shahriar
Jan 30, 2018

The number of squares is 1 2 + + n 2 1^2 + \ldots + n^2 , for a n × n n \times n square.

Putting n = 10 n = 10

1 2 + 2 2 + 3 2 + + 1 0 2 = 10 ( 11 ) ( 21 ) 6 = 385 1^2 + 2^2 + 3^2 + \ldots + 10^2 = \dfrac{10(11)(21)}{6} = \boxed{385}

Note i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 \quad \sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}

Jerry McKenzie - 3 years, 4 months ago

It would have been better for the solvers to have figured out the pattern themselves. You made the problem too easy: Level 0.

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