Math Gold Medalist

Lor

2023 AMC 10A

Problem 20

Each square in a $3\times3$ grid of squares is colored red, white, blue, or green so that every $2\times2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?

[asy] size(8cm); pen grey1, grey2, grey3; grey1 = RGB(211, 211, 211); grey2 = RGB(173, 173, 173); grey3 = RGB(138, 138, 138); for(int i = 0; i < 4; ++i) { draw((i, 0)--(i, 3)); draw((0, i)--(3, i)); } filldraw((5, 3)--(6, 3)--(6, 2)--(5, 2)--cycle, grey1); label('B', (5.5, 2.5)); filldraw((6, 3)--(7, 3)--(7, 2)--(6, 2)--cycle, grey2); label('R', (6.5, 2.5)); filldraw((7, 3)--(8, 3)--(8, 2)--(7, 2)--cycle, grey1); label('B', (7.5, 2.5)); filldraw((5, 2)--(6, 2)--(6, 1)--(5, 1)--cycle, grey3); label('G', (5.5, 1.5)); filldraw((6, 2)--(7, 2)--(7, 1)--(6, 1)--cycle, white); label('W', (6.5, 1.5)); filldraw((7, 2)--(8, 2)--(8, 1)--(7, 1)--cycle, grey3); label('G', (7.5, 1.5)); filldraw((5, 1)--(6, 1)--(6, 0)--(5, 0)--cycle, grey2); label('R', (5.5, 0.5)); filldraw((6, 1)--(7, 1)--(7, 0)--(6, 0)--cycle, grey1); label('B', (6.5, 0.5)); filldraw((7, 1)--(8, 1)--(8, 0)--(7, 0)--cycle, grey2); label('R', (7.5, 0.5)); [/asy]

$\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }96$

Counting

Casework

Priority: The most Limited Case

Consider first 2by2 and then first 2 cells of 3rd column

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   Solution