Question

下图显示一个点状网格，由 $8$ 格宽、 $3$ 格高的 $1'' \times 1''$ 正方形组成。Carl 沿一些小正方形的边放置 $1$ 英寸牙签，形成一条不自交的闭合回路。格子中的数字表示该正方形有多少条边要被牙签覆盖；若格子中没有数字，则允许任意数量的牙签。Carl 有多少种放置牙签的方法？

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?

$130$

$144$

$146$

$162$

$196$

答案：C

知识点：双射分类讨论系统列举

难度评级：2600

小提示：

中间一行的每个方格必须恰好接触一根牙签；两端折返的列确定后，每个内部方格的那根牙签独立地放在它的上边或下边。

Each middle-row cell must touch exactly one toothpick; after the two turnaround columns are fixed, each interior cell’s toothpick is independently above or below it

大提示：

穿过中间的回路会跨越全部 $8$ 列、前 $7$ 列、后 $7$ 列，或中间 $6$ 列；内部各列可以独立地向上或向下弯折

A loop crossing the middle uses all $8,$ the first $7,$ the last $7,$ or the middle $6$ columns; its interior columns independently bend above or below

解答：

中间一行的每个方格必须恰好接触一根牙签。先考虑从中间条带一侧穿到另一侧的回路。回路可以跨越全部 $8$ 列、前 $7$ 列、后 $7$ 列，或中间 $6$ 列；更窄的跨度会使外侧的一个中间方格没有接触牙签。固定两个端点后，每个内部的中间方格都可以独立选择把唯一一根牙签放在上边或下边，其余不自交回路随即唯一确定。因此四种情况分别给出 $2^6,2^5,2^5$ ，和 $2^4$ 条回路。另有恰好两条不穿过中间条带的回路：完全沿顶部或完全沿底部延伸的水平长方形。所以总数为 $2^6+2^5+2^5+2^4+2$ $=64+32+32+16+2$ $=146$ 。所以正确答案是 C。

Each middle-row cell must touch exactly one toothpick. First consider loops that pass from one side of the middle strip to the other. The loop can span all $8$ columns, the first $7,$ the last $7,$ or the middle $6;$ a narrower span would leave an outer middle cell untouched. Once the two ends are fixed, each interior middle cell independently has its one toothpick on its top or bottom side, and the rest of the non-self-intersecting loop is forced. The four cases therefore contribute $2^6,2^5,2^5,$ and $2^4$ loops. There are also exactly two loops that do not cross the middle strip: the horizontal rectangle running entirely along the top or entirely along the bottom. Hence the total is $2^6+2^5+2^5+2^4+2$ $=64+32+32+16+2$ $=146.$ Thus, C is the correct answer.

2024 AMC 10A 第 25 题

答案：C

难度评级：2600

小提示：

大提示：

解答：

其他年份的第 25 题