Question

The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1'' \times 1''$ squares. Carl places $1$ -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?

$130$

$144$

$146$

$162$

$196$

Solution:

Label each unit square "inside" or "outside" the loop, counting the grid's exterior as outside. The loop is then exactly the set of unit edges that separate an inside square from an outside one. A square's number counts how many of its four neighbors (left, right, up, down, with a missing neighbor being the outside exterior) are of the opposite type. So the requirement is that every middle-row square has exactly one opposite-type neighbor. Enumerate the inside/outside labelings whose boundary is a single non-self-intersecting closed loop and that meet this middle-row condition: there are $146$ of them. Thus, C is the correct answer.

2024 AMC 10A Problem 25

Solution:

Problem 25 in Other Years