A probability problem by Matin Naseri

Probability Level 2

How many square's A 12×12 \text{12×12} chess board include??


The answer is 650.

Matin Naseri by Matin Naseri , 17, Afghanistan

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3 solutions

Matin Naseri
Feb 1, 2018

( n ) ( n + 1 = A ) ( n + A = m ) 6 \frac{(n)(n+1=A)(n+A=m)}{6}

( 12 ) ( 12 + 1 = 13 ) ( 12 + 13 = 25 ) 6 \frac{(12)(12+1=13)(12+13=25)}{6} = 3900 6 \frac{3900}{6} = 650 \text{650}

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Mohammad Farhat
Oct 15, 2018

Another solution that I did not see is:

1 2 2 + 1 1 2 + 1 0 2 + 9 2 + ⋯ + 1 2 = 650 \displaystyle 12^2 + 11^2 + 10^2 + 9^2 + \cdots + 1^2 =650

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Munem Shahriar
Feb 1, 2018

The number squares is n ( n + 1 ) ( 2 n + 1 ) 6 \dfrac{n(n+1)(2n+1)}{6} , for a n × n n \times n square.

n ( n + 1 ) ( 2 n + 1 ) 6 = 12 ( 13 ) ( 25 ) 6 = 3900 6 = 650 \large \frac{n(n+1)(2n+1)}{6} = \frac{12(13)(25)}{6} = \frac{3900}{6} = \boxed{650}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Munem!

You are have a typo.

n ( n + 1 ) ( 2 n + 1 ) 6 = 12 ( 13 ) ( 25 ) 6 = 3900 60 = 650 \large \frac{n(n+1)(2n+1)}{6} = \frac{12(13)(25)}{6} = \frac{3900}{60} = \boxed{650}

It must be:

n ( n + 1 ) ( 2 n + 1 ) 6 = 12 ( 13 ) ( 25 ) 6 = 3900 6 = 650 \large \frac{n(n+1)(2n+1)}{6} = \frac{12(13)(25)}{6} = \frac{3900}{6} = \boxed{650}

Matin Naseri - 3 years, 4 months ago

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Thanks. :)

Munem Shahriar - 3 years, 4 months ago

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No problem.

Matin Naseri - 3 years, 4 months ago

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