Question

Two of the squares of a $7\times7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?

Answer

Answer: 300

Concepts:Burnside’s Lemma combinations rotational symmetry

Difficulty rating: 2270

Small Hint:

Average the numbers of two-square colorings fixed by the four rotations

Big Hint:

Only a half-turn can fix a nontrivial pair of squares

Solution:

Under the identity rotation, all $\binom{49}{2}=1176$ pairs are fixed. A quarter-turn or three-quarter-turn has only orbits of sizes $1$ and $4,$ so it fixes no two-square set. A half-turn fixes exactly the $24$ pairs of squares opposite one another across the center. Burnside’s Lemma therefore gives $\frac{1176+0+24+0}{4}=300$ inequivalent colorings.

1996 AIME Problem 7

Answer: 300

Difficulty rating: 2270

Small Hint:

Big Hint:

Solution:

Problem 7 in Other Years