2021 AMC 12B Problems/Problem 19

Problem

Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{1}{12}$ . What is the least possible number of faces on the two dice combined?

$\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20$

Solution 1

Suppose the dice have $a$ and $b$ faces, and WLOG $a\geq{b}$ . Since each die has at least $6$ faces, there will always be $6$ ways to sum to $7$ . As a result, there must be $\tfrac{4}{3}\cdot6=8$ ways to sum to $10$ . There are at most nine distinct ways to get a sum of $10$ , which are possible whenever $a,b\geq{9}$ . To achieve exactly eight ways, $b$ must have $8$ faces, and $a\geq9$ . Let $n$ be the number of ways to obtain a sum of $12$ , then $\tfrac{n}{8a}=\tfrac{1}{12}\implies n=\tfrac{2}{3}a$ . Since $b=8$ , $n\leq8\implies a\leq{12}$ . In addition to $3\mid{a}$ , we only have to test $a=9,12$ , of which both work. Taking the smaller one, our answer becomes $a+b=9+8=\boxed{\textbf{(B)}\ 17}$ .

Solution 2

Suppose the dice have $a$ and $b$ faces, and WLOG $a\geq{b}$ . Note that if $a+b=12$ since they are both $6$ , there is one way to make $12$ , and incrementing $a$ or $b$ by one will add another way. This gives us the probability of making a 12 as $\frac{a+b-11}{ab}=\frac{1}{12}$ Cross-multiplying, we get $12a+12b-132=ab$ Simon's Favorite Factoring Trick now gives $(a-12)(b-12)=12$ This narrows the possibilities down to 3 ordered pairs of $(a,b)$ , which are $(13,24)$ , $(6,10)$ , and $(8,9)$ . We can obviously ignore the first pair and test the next two straightforwardly. The last pair yields the answer: $\frac{6}{72}=\frac{3}{4}\left(\frac{9+8-9}{72}\right)$ The answer is then $a+b=8+9=\boxed{\textbf{(B)}\ 17}$ .

~Hyprox1413

Solution 3 (Logic)

Notice that $\begin{align*} 7&=1+6 \\ &\ \ \ \ \ \ \ \ \ \vdots \\ &=6+1 \end{align*}$ are the only cases that could possibly form 7. In another words, regardless of the number of faces for each dice, 6 is the number of cases that could form a rolling sum of 7.

Because the value of the total combination for both are the same, we could infer that the number of cases of having a rolling sum of 10 is $\frac{4}{3}\cdot6=8$ . With the number 8, we could also deduce that one dice has 8 sides and the other has at least 9 sides. Thence trial and error could be utilized.

Since 9 is the next smallest number, the case could be tested. $\begin{align*} 12&=3+9 \\ &=4+8 \\ &\ \ \ \ \ \ \ \ \ \vdots \\ &=8+4 \end{align*}$ $\frac{6}{8\cdot9}=\frac{1}{12}$ is also true. Therefore, $8+9=\boxed{\textbf{(B) }17}$

~MaPhyCom

Video Solution

https://youtu.be/xGp5yQ5Bshs

~MathProblemSolvingSkills

Video Solution by OmegaLearn (Using Probability)

https://youtu.be/geEDrsV5Glw

~ pi_is_3.14

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

Solution 4:

Start with the minimum number of answers, which is A:16. For two dice with at least six faces, the possible numbers of two faces are (6, 10), (7, 9), and (8, 8). We can easily rule out (7,9) and (8,8) because their products (63 and 64, respectively) are not multiples of 12. Then, the probability of having a sum of 12 from these two dice is not possible to be 1/12. (6,10) does not satisfy the requirement that the probability of rolling a sum of 7 is 3/4 of the probability of rolling a sum of 10. So A is not possible.

Next, try B:17. The possible numbers of two faces are (6,11), (7, 10), and (8,9). Similarly, we can rule out (6,11) and (7,10) easily because their products (66 and 70, respectively) are not multiples of 12. Check (8,9): the probability of having a sum of 12 is 6/12 = 1/12. The combinations with a sum of 12 are: 3-9, 4-8, 5-7, 6-6, 7-5, 8-4, totaling six pairs. There is no 9-3 combination because only one die can have nine faces. The combinations with a sum of 10 are: 1-9, 2-8, 3-7, 4-6, 5-5, 6-4, 7-3, 8-2, 8 pairs. Again, no 9-1 because there is only one nine-face die. The combinations of a sum of 7 are: 1-6, 2-5, 3-4, 4-3, 5-2, and 6-1, 6 pairs. So, the probability of rolling a sum of 7 is 3/4 of the probability of rolling a sum of 10. So B is possible.

Since the question asks for the minimum possible number, B is the answer.

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