2023 AMC 10A Problem 20

Each square in a

3 \times 3

grid of squares is colored red, white, blue, or green so that every

2 \times 2

square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?

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Solution

Label the cells row by row as

a,

b,

c;

d,

e,

f;

g,

h,

i.

The top-left block

a, b, d, e

is a permutation of the four colors, so

4! = 24

ways. The block

\{b, c, e, f\}

is also all four colors, and

b, e

are fixed, so

\{c, f\}

is the remaining two in some order:

2

ways. Same story for

\{g, h\},

the two colors apart from

d, e,

another

2

ways. That leaves

i,

forced to whatever color is missing from

\{e, f, h\},

and that only works when

f \ne h.

Of the

2 \cdot 2 = 4

order combinations, exactly one has

f = h,

3

survive. The total is

24 \cdot 3 = 72.

Therefore, the answer is D.