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2012 AMC 10A Problem 14

2012 AMC 10A Problem 14AMC 10 Step by Step

Name: Date: September 13, 2026

Paths & Grids · difficulty 2 of 5 · about 3 min · position P11-15

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?

A)
$480$
B)
$481$
C)
$482$
D)
$483$
E)
$484$

Answer

Key insight

Color depends only on the parity of row + column; with corners black, black squares are those with an even coordinate sum, one more than half of 961.

Solution

Number the rows and columns $1$ through $31$ . Alternating colors along every row and column means the color of square $(r, c)$ depends only on whether $r + c$ is even or odd. The corners $(1,1)$ , $(1,31)$ , $(31,1)$ , $(31,31)$ all have even sums and are black, so black means $r + c$ even.

Count squares with $r + c$ even. Row $1$ (odd) has black squares in the $16$ odd columns; row $2$ (even) has black squares in the $15$ even columns. The pattern alternates $16, 15, 16, 15, \dots$ over the $31$ rows: $16$ odd rows contribute $16$ each and $15$ even rows contribute $15$ each:
$16 \cdot 16 + 15 \cdot 15 = 256 + 225 = 481.$

The answer is $\boxed{\textbf{(B)}\ 481}$ .

Why this works

A checkerboard is a parity coloring. On an even-sided board the two colors split evenly, but on an odd-sided board the color of the corners gets the extra square: the $961$ squares split as $481$ black and $480$ red. Whenever a board has odd dimensions, expect a one-square imbalance in favor of the corner color.

Alternative approach

Pair squares along each row: each row of $31$ has $15$ red-black pairs plus one leftover of the color the row starts with. Odd rows start black, even rows start red, so leftovers give $16$ extra blacks and $15$ extra reds. Total black $= 15 \cdot 31 + 16 = 481$ .

The trap

Answering 961/2 rounded down (480), forgetting that an odd-by-odd board with black corners has one extra black square.

Common mistakes

Answering 961/2 rounded down (480), forgetting that an odd-by-odd board with black corners has one extra black square.
Miscounting the number of odd numbers from $1$ to $31$ as $15$ instead of $16$ .

Techniques

Set up the equation/formula and compute; no special trick needed · Use an invariant, parity, or coloring argument