Difference between revisions of "2021 Fall AMC 10A Problems/Problem 18"
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| + | ==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] | [asy] | ||
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[/asy] | [/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> | ||
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| + | ==Risky, but Quick Solution== | ||
| + | The top right box has <math>4</math> choices and the top left box has <math>3</math> choices. Thus, it is reasonable to assume that the answer is a multiple of <math>12</math>. We know that the answer will not be too small or too large, so the answer is <math>\boxed{\textbf{(C)}\ 84}</math> | ||
Revision as of 18:43, 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]
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 
