Minimum Dice Sum!

Probability Level 5

$n$ six-sided dice are rolled. It is given that the probability of obtaining the sum of $2014$ is non-zero and is the same as the probability of obtaining a sum of $S$ . What is the minimum value of $S$ ?

This problem is from the AHSME.

This problem is from the set "Olympiads and Contests Around the World -3". You can see the rest of the problems here .

The answer is 338.

2 solutions

Lam Nguyen
Feb 3, 2015

There are n dices, thus the sum varies from n to 6n We see that the possibility of getting n + i is the same as getting 6n - i Thus we find n and i so that: i + n = 2014 and 6n - i is the smallest Substituting -i with n - 2014 yields We need to find the minimum value of 7n - 2014 Because 2014 is a constant, we need to find the smallest possible n Because n<=2014<=6n Thus 336<=n<=2014 Substituting n = 336 gives S = 338

Parthasarathi Khirwadkar
Nov 29, 2014

The minimum number of dice required to have a sum of 2014 will be 336. Now there are 2 ways to get a sum of 2014:

(i) 335 show 6 and one of the dice shows 4 (ii)334 show 6 and two of the dice show 5

If 333 dice show 6 then the sum we need to get out of remaining 3 dices is 16 such that none show 6(as that would be same as above) and we can only get a sum of 15 in this way, so there are only 2 ways.

Now to get minimum sum, we can take the minimum value on each dice. In this way we will have one way to get 336 and 337 and two ways to get 338 (can be proven in same way as above).

Minimum Dice Sum!

The answer is 338.

2 solutions

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