Math Gold Medalist

Lor

2023 AIME II

Problem 10

Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$\[\begin{array}{|c|c|c|c|c|c|} \hline \,1\, & \,3\, & \,5\, & \,7\, & \,9\, & 11 \\ \hline \,2\, & \,4\, & \,6\, & \,8\, & 10 & 12 \\ \hline \end{array}\]

Modular Arithmetic

Double Counting

If we consider 1 to 12 mod 3 then #0=4, #1=4, #2=4

Each column is 01 or 02 or 03 (0 and 1 can be switched)

Prove that number of 01 columns = 2

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   Solution