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Slitherlink – Loop the Loop点阵网格闭合回路

作者：

radmin

在

https://artofproblemsolving.com/wiki/index.php?title=2024_AMC_10A_Problems/Problem_25

The figure below shows a dotted grid 8  cells wide and  3  cells tall consisting of  1‘‘*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?

[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]
$\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$

https://artofproblemsolving.com/wiki/index.php/2026_AIME_I_Problems/Problem_15

Let $a, b,$ and $n$ be positive integers with both $a$ and $b$ greater than or equal to $2$ and less than or equal to $2n.{}$ Define an $a \times b$ cell loop in a $2n \times 2n$ grid of cells to be the $2a + 2b – 4$ cells that surround an $(a – 2) \times (b – 2)$ (possibly empty) rectangle of cells in the grid. For example, the following diagram shows a way to partition a $6 \times 6$ grid of cells into $4$ cell loops.

Find the number of ways to partition a $10\times10$ grid of cells into $5$ cell loops so that every cell of the grid belongs to exactly one cell loop.

[asy] usepackage(“mathptmx”); size(6cm); dotfactor = 30; fill((0, 2)--(0, 6)--(4, 6)--(4, 2)--cycle, royalblue); fill((1, 3)--(1, 5)--(3, 5)--(3, 3)--cycle, lightred); fill((4, 0)--(4, 6)--(6, 6)--(6, 0)--cycle, lightyellow); draw((1, 1)--(3, 1)^^(5, 1)--(5, 5)^^(1, 3)--(1, 5)--(3, 5)--(3, 3)--cycle^^(0, 0)--(0, 6)--(6, 6)--(6, 0)--cycle^^(0, 2)--(4, 2)^^(4, 0)--(4, 6), linewidth(4)); dot((2, 4)); for (int i = 1; i <= 5; i += 1) { draw((0, i)--(6, i)^^(i, 0)--(i, 6)); } [/asy]

Q3 https://artofproblemsolving.com/wiki/index.php/2023_AIME_I_Problems/Problem_14

The following analog clock has two hands that can move independently of each other.

Initially, both hands point to the number $12$. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock face while the other hand does not move.

Let $N$ be the number of sequences of $144$ hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the $144$ movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by $1000$.

[asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 1; i < 13; ++i) { label("\small" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy]

https://artofproblemsolving.com/wiki/index.php/2017_AIME_II_Problems/Problem_11

Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).

Q1 146

Q2 83

Q3 608

Q4 544

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