Difference between revisions of "2024 AMC 10A Problems/Problem 25"
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| − | + | ==Problem== | |
| + | The figure below shows a dotted grid <math>8</math> cells wide and <math>3</math> cells tall consisting of <math>1''\times1''</math> squares. Carl places <math>1</math>-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? | ||
| + | <asy> | ||
| + | size(6cm); | ||
| + | for (int i=0; i<9; ++i) { | ||
| + | draw((i,0)--(i,3),dotted); | ||
| + | } | ||
| + | for (int i=0; i<4; ++i){ | ||
| + | draw((0,i)--(8,i),dotted); | ||
| + | } | ||
| + | for (int i=0; i<8; ++i) { | ||
| + | for (int j=0; j<3; ++j) { | ||
| + | if (j==1) { | ||
| + | label("1",(i+0.5,1.5)); | ||
| + | }}} | ||
| + | </asy> | ||
| + | <math>\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196</math> | ||
| + | ==Solution== | ||
Revision as of 16:24, 8 November 2024
Problem
The figure below shows a dotted grid 
cells wide and 
cells tall consisting of 
squares. Carl places 
-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks?
![[asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy]](http://latex.artofproblemsolving.com/9/d/1/9d176dfd41e3e8e875c6b5d84fdb4f5e4f7d43a6.png)
