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2023 AMC 12A Problems/Problem 21

The following problem is from both the 2023 AMC 10A #25 and 2023 AMC 12A #21, so both problems redirect to this page.

Problem

If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?

$\textbf{(A) } \frac{7}{22} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{3}{8} \qquad \textbf{(D) } \frac{5}{12} \qquad \textbf{(E) } \frac{1}{2}$

1 minute solution by MegaMath

https://www.youtube.com/watch?v=dxYw1wYHid4&t=12s

Solution 1

To find the total amount of vertices we first find the amount of edges, and that is $\frac{20 \times 3}{2}$. Next, to find the amount of vertices we can use Euler's characteristic, $V - E + F = 2$, and therefore the amount of vertices is $12$

So there are $P(12,3) = 1320$ ways to choose 3 distinct points.

Now, the furthest distance we can get from one point to another point in an icosahedron is 3. Which gives us a range of $1 \leq d(Q, R), d(R, S) \leq 3$

With some case work, we get two cases:

Case 1: $d(Q, R) = 3; d(R, S) = 1, 2$

Since we have only one way to choose Q, that is, the opposite point from R, we have one option for Q and any of the other points could work for S.

Then, we get $12 \times 1 \times 10 = 120$ (ways to choose R × ways to choose Q × ways to choose S)

Case 2: $d(Q, R) = 2; d(R, S) = 1$

We can visualize the icosahedron as 4 rows, first row with 1 vertex, second row with 5 vertices, third row with 5 vertices and fourth row with 1 vertex. We set R as the one vertex on the first row, and we have 12 options for R. Then, Q can be any of the 5 points on the third row and finally S can be one of the 5 points on the second row.

Therefore, we have $12 \times 5 \times 5 = 300$ (ways to choose R × ways to choose Q × ways to choose S)

Hence, $P(d(Q, R)>d(R, S)) = \frac{120+300}{1320} = \boxed{\textbf{(A) } \frac{7}{22}}$

~lptoggled, edited by ESAOPS and trevian1

Solution 2 (Cheese + Actual way)

In total, there are $\binom{12}{3}=220$ ways to select the points. However, if we look at the denominators of $B,C,D$, they are $3,8,12$ which are not divisors of $220$. Also $\frac{1}{2}$ is impossible as cases like $d(Q, R) = d(R, S)$ exist. The only answer choice left is $\boxed{\textbf{(A) } \frac{7}{22}}$

Note: this cheese is actually wrong because the total number of ways to select the points is actually $12 \times 11 \times 10 = 1320$ as order matters, so all denominators are possible. Rather, you can arrive at the same conclusion by fixing R WLOG, leading to $11 \times 10 = 110$ ways in total, which works for the original cheese. ~awesomeguy856

(Actual way)

Fix an arbitrary point, to select the rest $2$ points, there are $11\cdot 10 =110$ ways. To make $d(Q, R)=d(R, S), d=1/2$. Which means there are in total $2\cdot 5 \cdot 4 = 40$ ways to make the distance the same. $\frac{1}{2}(1-\frac{40}{110})= \boxed{\textbf{(A) } \frac{7}{22}}$ ~bluesoul

Solution 3

We can imagine the icosahedron as having 4 layers. 1 vertex at the top, 5 vertices below connected to the top vertex, 5 vertices below that which are 2 edges away from the top vertex, and one vertex at the bottom that is 3 edges away. WLOG because the icosahedron is symmetric around all vertices, we can say that R is the vertex at the top. So now, we just need to find the probability that S is on a layer closer to the top than Q. We can do casework on the layer S is on to get \[\frac{5}{11} \cdot \frac{6}{10} + \frac{5}{11} \cdot \frac{1}{10} = \frac{35}{110} = \frac{7}{22}\] So the answer is $\boxed{\textbf{(A) }\frac{7}{22}}$. -awesomeparrot

Solution 4

We can actually see that the probability that $d(Q, R) > d(R, S)$ is the exact same as $d(Q, R) < d(R, S)$ because $d(Q, R)$ and $d(R, S)$ have no difference. (In other words, we can just swap Q and S, meaning that can be called the same probability-wise.) Therefore, we want to find the probability that $d(Q, R) = d(R, S)$.

WLOG, we can rotate the icosahedron so that R is the top of the icosahedron. Then we can divide this into 2 cases:

1. They are on the second layer

There are 5 ways to put one point, and 4 ways to put the other point such that $d(Q, R) = d(R, S) = 1$. So, there are $5 \cdot 4 = 20$ ways to put them on the second layer.

2. They are on the third layer

There are 5 ways to put one point, and 4 ways to put the other point such that $d(Q, R) = d(R, S) = 2$. So, there are $5 \cdot 4 = 20$ ways to put them on the third layer.

The total number of ways to choose P and S are $11 \cdot 10 = 110$ (because there are 12 vertices), so the probability that $d(Q, R) = d(R, S)$ is $\frac{20+20}{110} = \frac{4}{11}$.

Therefore, the probability that $d(Q, R) > d(R, S)$ is $\frac{1 - \frac{4}{11}}{2} = \boxed{\textbf{(A) }\frac{7}{22}}$

~Ethanzhang1001

Solution 5

We know that there are $20$ faces. Each of those faces has $3$ borders (since each is a triangle), and each edge is used as a border twice (for each face on either side). Thus, there are $\dfrac{20\cdot3}2=30$ edges.

By Euler's formula, which states that $v-e+f=2$ for all convex polyhedra, we know that there are $2-f+e=12$ vertices.

The answer can be counted by first counting the number of possible paths that will yield $d(Q, R) > d(R, S)$ and dividing it by $12\cdot11\cdot10$ (or $\dbinom{12}3$, depending on the approach). In either case, one will end up dividing by $11$ somewhere in the denominator. We can then hope that there will be no factor of $11$ in the numerator (which would cancel the $11$ in the denominator out), and answer the only option that has an $11$ in the denominator: $\boxed{\textbf{(A) }\frac{7}{22}}$.

~Technodoggo

Additional note by "Fruitz": Note that one can eliminate $1/2$ by symmetry if you swap the ineq sign.

Another note by "andliu766": A shorter way to find the number of vertices and edges is to use the fact that the MAA logo is an icosahedron. :)

Solution 6 (Case Work)

2023AMC12AP21.png

WLOG, let R be at the top-most vertex of the icosahedron. There are $2$ cases where $d(Q, R) > d(R, S)$.

Case 1: $Q$ is at the bottom-most vertex

If $Q$ is at the bottom-most vertex, no matter where $S$ is, $d(Q, R) > d(R, S)$. The probability that $Q$ is at the bottom-most vertex is $\frac{1}{11}$

Case 2: $Q$ is at the second layer

If $Q$ is at the second layer, $S$ must be at the first layer, for $d(Q, R) > d(R, S)$ to be true. The probability that $Q$ is at the second layer, and $S$ is at the first layer is $\frac{5}{11} \cdot \frac{5}{10} = \frac{5}{22}$

\[\frac{1}{11} + \frac{5}{22} = \boxed{\textbf{(A) }\frac{7}{22}}\]

~isabelchen

Solution 7 (efficient)

Since the icosahedron is symmetric polyhedron, we can rotate it so that R is on the topmost vertex. Since Q and S basically the same, we can first count the probability that $d(Q,R) = d(R,S)$.

$\mathfrak{Case} \ \mathfrak{1}: d(Q,R) = d(R,S) = 1$

There are 5 points $P$ such that $d(Q,P) = 1$. There is $5 \times 4 = \boxed{20}$ ways to choose Q and S in this case.

$\mathfrak{Case} \ \mathfrak{2}: d(Q,R) = d(R,S) = 2$

There are 5 points $P$ such that $d(Q,P) = 2$. There is $5 \times 4 = \boxed{20}$ ways to choose Q and S in this case.

$\mathfrak{Case} \ \mathfrak{3}: d(Q,R) = d(R,S) = 3$

There is 1 point $P$ such that $d(Q,P) = 3$. There is $1 \times 0 = \boxed{0}$ ways to choose Q and S in this case.

$\mathfrak{Final} \ \mathfrak{solution}$

There are 11 points $P$ that are distinct from R. There is $11 \times 10 = \boxed{110}$ ways to choose Q and S. There is $20 + 20 + 0 = \boxed{40}$ ways to choose Q and S such that $d(Q,R) = d(R,S)$. There is $\frac{110-40}{2} = 35$ ways to choose Q and S such that $d(Q, R) > d(R, S)$. The probability that $d(Q, R) > d(R, S)$ is therefore $\frac{35}{110} = \frac{7}{22}$ which corresponds to answer choice $\boxed{A}$

~~afly

Solution 8 (Dodecahedron Schlegel Diagram)

A-plane-representation-of-the-dodecahedron.png

Instead of thinking this problem as vertices of a icosahedron, think of it as faces of a dodecahedron. Our middle face is $R$ and we first choose $Q$ and $S$ to be the faces with distance $1$ from the middle face. There are $\binom{5}{2}$ ways of doing that. Then we choose $Q$ and $S$ to be the faces with distance $2$ from the middle face. There are $\binom{5}{2}$ ways of doing that. Our answer is therefore $\frac{1}{2}\left(1-\frac{2\binom{5}{2}}{\binom{11}{2}}\right)=\boxed{\textbf{(A) }\frac{7}{22}}$

~lopkiloinm

Solution 9 (Symmetry)

Note that the vertices of an icosahedron are in a $1-5-5-1$ configuration.

We will denote the probability of event $A$ happening as $P(A)$

We should also note that $P(d(Q, R) > d(R, S)) + P(d(Q, R) < d(R, S)) + P(d(Q, R) = d(R, S)) = 1$

Also, $P(d(Q, R) > d(R, S)) = P(d(Q, R) < d(R, S))$ since $Q$ and $S$ are basically the same.

Therefore, what we are trying to find is just $\frac{1-P(d(Q, R) = d(R, S))}{2}$ which can easily be computed when we fix $R$ at the top vertex in the $1-5-5-1$ structure.

Since all the points are distinct (stated in problem), the conditions are only satisfied where $Q$ and $S$ are at $5$s of the $1-5-5-1$ structure.

$P(d(Q, R) = d(R, S)) = 2 \cdot \frac{5}{11} \cdot \frac{4}{10} = \frac{4}{11}$ since we have to choose one of the $5$ out of $11$ possible vertices and another one of $4$ out of $10$ two times.

And the answer is $\frac{1-\frac{4}{11}}{2} = \boxed{ \textbf{(A) } \frac{7}{22}}$

~jjaamm

Video Solution

https://youtu.be/4PSrAbrjKVg?si=G24hhBrhUPio9SZD

~MathProblemSolvingSkills.com


Video Solution by OmegaLearn

https://youtu.be/Wc6PFNq5PAM

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=1yLvQ0F5Z-E

Video Solution by epicbird08

https://youtu.be/s_6q0C0z6Ug

~EpicBird08

Vide Solution by SpreadTheMathLove(Casework and Complementary)

https://www.youtube.com/watch?v=4FEwxwgbliQ

Video Solution

https://youtu.be/bS3tle-jP4g

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution by TheBeautyofMath

https://youtu.be/ZvrlZ2NoH8s

~IceMatrix

See also

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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