Difference between revisions of "2021 Fall AMC 10A Problems/Problem 18"
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==Problem== | ==Problem== | ||
A farmer's rectangular field is partitioned into <math>2</math> by <math>2</math> grid of <math>4</math> 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? | A farmer's rectangular field is partitioned into <math>2</math> by <math>2</math> grid of <math>4</math> 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((0,0)--(100,0)--(100,50)--(0,50)--cycle); | ||
draw((50,0)--(50,50)); | draw((50,0)--(50,50)); | ||
draw((0,25)--(100,25)); | draw((0,25)--(100,25)); | ||
| − | + | </asy> | |
<math>\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 84 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 144</math> | <math>\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 84 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 144</math> | ||
Revision as of 18:57, 22 November 2021
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]](http://latex.artofproblemsolving.com/a/4/9/a4955e6b2cb243b0a518bec889e6971862d5599c.png)
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 
- Aidensharp
