Playing the Game

Algebra Level 2

Two chess masters played 11 games of chess against each other, each taking one move longer than the last one. The 4 th 4^\text{th} game took 28 moves.

How many total moves were used in the 11 games?


The answer is 330.

Denton Young by Denton Young , 47, USA

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2 solutions

Hung Woei Neoh
Jun 1, 2016

The number of moves used in each chess game forms an arithmetic progression where

n = 11 , d = 1 , T 4 = 28 n=11,\;d=1,\;T_4 = 28

We need to find the first term a a :

a + ( 4 − 1 ) ( 1 ) = 28 a + 3 = 28 a = 25 a+(4-1)(1) = 28\\ a+3=28\\ a=25

The total moves

= S 11 = 11 2 ( 2 ( 25 ) + ( 11 − 1 ) ( 1 ) ) = 11 2 ( 50 + 10 ) = 11 2 ( 60 ) = 11 ( 30 ) = 330 =S_{11}\\ =\dfrac{11}{2}(2(25) + (11-1)(1))\\ =\dfrac{11}{2}(50+10)\\ =\dfrac{11}{2}(60)\\ =11(30)\\ =\boxed{330}

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Denton Young
Jun 1, 2016

If the 4th game took 28 moves, the 6th game must have taken 30.

Since the series is symmetric around the 6th game, the total number of moves used is 30 * 11 = 330 moves.

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Moderator note:

Great approach of figuring out the middle term to find the AP sum.

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