Question

The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ 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?

A dotted grid 8 cells wide and 3 cells tall, with a 1 written in each cell of the middle row.

$130$

$144$

$146$

$162$

$196$

Solution:

The toothpicks form a single simple closed curve. The constraint forces each of the eight middle-row cells to have exactly one of its four sides used, which severely limits how the loop threads through the grid: for each middle cell the loop must contribute one edge (a top, bottom, left, or right side), and consecutive choices must join up into one non-self-intersecting closed curve.

Working left to right and tracking how the loop's upper and lower portions enter and leave each column (equivalently, a transfer-matrix/casework count over the $8$ columns) enumerates all admissible loops. Carrying out this casework gives $146$ valid configurations.

Thus, the correct answer is C.

2024 AMC 12A Problem 22

Solution:

Problem 22 in Other Years