2012 AMC 10A Problem 14

Question

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?

480

481

482

483

484

Answer: B

Concepts:basic counting parity pattern recognition

Difficulty rating: 1020

Small hint:

Rows alternate between

16

and

15

black squares

Big hint:

There are

16

rows of one type and

15

of the other

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.

2012 AMC 10A Problem 14

Answer: B

Difficulty rating: 1020

Small hint:

Big hint:

Solution:

Problem 14 in other years