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Difference between revisions of "1994 AHSME Problems/Problem 30"

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<math> \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341 </math>
 
<math> \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341 </math>
 
==Solution==
 
==Solution==
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Given that there are <math>n</math> dice, we know that the chance of rolling a sum of <math>1994</math> is the same as that of rolling a sum of <math>7n - 1994</math>. This is because there exists a bijection between the set <math>A</math> of dice rolls that sum to <math>1994</math> and the set <math>B</math> of dice rolls that sum to <math>7n - 1994</math>. In other words, for every ordered n-tuple <math>(a_1, a_2, \dots, a_n) \in A</math> such that <math>a_i \in \{1,2,3,4,5,6\}</math> for all valid <math>i</math> and that <math>\sum^{1994}_{i=0} a_i = 1994</math>, there is a unique ordered n-tuple <math>(b_1, b_2, \dots, b_n) \in B</math> where <math>b_i = 7 - a_i</math> for all valid <math>i</math>, whose sum is indeed <math>7n - 1994</math>.
 
  
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As such, minimizing <math>S</math> is the same as minimizing <math>n</math>. The minimum value of <math>n</math> is <math>\left\lceil \frac{1994}{6} \right\rceil = 333</math>. Hence, <math>S = 7n - 1994 = 337</math> <math>\textbf{(C)}</math>.
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Let <math>d_i</math> be the number on the <math>i</math>th die.  There is a symmetry where we can replace each die's result with <math>d_i' = 7-d_i</math>.  Note that applying the symmetry twice we get back to where we started since <math>7-(7-d_i)=d_i</math>.  Under this symmetry the sum <math>S=\sum_{i=1}^n d_i</math> is replaced by <math>S' = \sum_{i=1}^n 7-d_i = 7n - S</math>.  As a result of this symmetry the probabilities of obtaing the sum <math>S</math> and the sum <math>S'</math> are equal because any combination of <math>d_i</math> which sum to <math>S</math> can be replaced with <math>d_i'</math> to get the sum <math>S'</math>, and conversely.  In other words, there is a bijection between the combinations of dice which sum to <math>S</math> and the combinations which sum to <math>S'</math>.
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The smallest non-zero probability of obtaining <math>S=1994</math> occurs when there are just enough dice for this sum to be possible, and any fewer dice would result in <math>S=1994</math> being impossible.  That happens when <math> n = \left\lceil \frac{1994}{6} \right\rceil = 333</math>. By the symmetry mentioned above, the probability of rolling <math>S=1994</math> is the same as the probability of rolling <math>S' = 333\cdot 7 - 1994 = 337</math>.  The answer is <math>\textbf{(C)}</math>.
  
 
==See Also==
 
==See Also==

Revision as of 03:06, 28 May 2021

Problem

When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is

$\textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341$

Solution

Let $d_i$ be the number on the $i$th die. There is a symmetry where we can replace each die's result with $d_i' = 7-d_i$. Note that applying the symmetry twice we get back to where we started since $7-(7-d_i)=d_i$. Under this symmetry the sum $S=\sum_{i=1}^n d_i$ is replaced by $S' = \sum_{i=1}^n 7-d_i = 7n - S$. As a result of this symmetry the probabilities of obtaing the sum $S$ and the sum $S'$ are equal because any combination of $d_i$ which sum to $S$ can be replaced with $d_i'$ to get the sum $S'$, and conversely. In other words, there is a bijection between the combinations of dice which sum to $S$ and the combinations which sum to $S'$.


The smallest non-zero probability of obtaining $S=1994$ occurs when there are just enough dice for this sum to be possible, and any fewer dice would result in $S=1994$ being impossible. That happens when $n = \left\lceil \frac{1994}{6} \right\rceil = 333$. By the symmetry mentioned above, the probability of rolling $S=1994$ is the same as the probability of rolling $S' = 333\cdot 7 - 1994 = 337$. The answer is $\textbf{(C)}$.

See Also

1994 AHSME (Problems • Answer Key • Resources)
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