2021 Fall AMC 10A Problems/Problem 18
Problem
A farmer's rectangular field is partitioned into 
by grid of 
rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
[asy]
draw((0,0)--(100,0)--(100,50)--(0,50)--cycle);
draw((50,0)--(50,50));
draw((0,25)--(100,25));
[/asy]
Risky, but Quick Solution
The top right box has choices and the top left box has 
choices. Thus, it is reasonable to assume that the answer is a multiple of 
. We know that the answer will not be too small or too large, so the answer is 
