2012 AMC 10A Problem 14

Problem 14 of 25IntermediateNumber Theory Counting & Probability

Chubby makes nonstandard checkerboards that have

31

squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?

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Solution

Note that there are

15

rows with

15

black tiles and

16

rows with

16

black tiles. This can be seen by observing that the first row has

16

black tiles, and all the other rows alternate with

15

and

16

tiles. Then, due to the alternating pattern, there will be a total of

16

rows with

16

tiles, and the other rows have

15

tiles. The total number of black squares is then

15^2 + 16^2 = 481.

Thus, B is the correct answer.

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