Question: Three dice, each with faces numbered 1 through 6, were tossed onto a game board. If one of the dice turned up 4, what was the sum of the numbers that turned up on all three dice?

1. The sum of two of the number that turned up was 10
2. The sum of two of the numbers that turned up was 11

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Solution and Explanation:

Approach Solution (1):

1. The sum of the two of the numbers that turned up was 10. The sum can be 4 + (5 + 5) = 14 or (4 + 6) + any > 10

Not sufficient

2. The sum of two of the numbers that turned up was 11: x + y = 11: The first dice turned up 4, so it can be neither x nor y (because the other one will be 7, so more than 6). Thus the sum is 4 + (x + y) = 15

Sufficient

Correct Option: B

Approach Solution (2):

Let there be three numbers D1, D2, and D3
D1 = 4

There are two possibilities for S1
(5,5, 4) and (4, 6, anything)

If D1 = 4 and D2 = 6 then D3 can be anything and we cannot find the sum
Hence S1 is insufficient

For S2, there is only one combination for making 11 i.e. (4, 5, 6)
Here we have two combinations

1. D1 = 4 D2 = 5 D3 = 6
2. D1 = 4 D2 = 6 D3 = 5

But since in both the cases the sum is 15… hence the statement 2 is sufficient

Correct Option: B

Approach Solution (3):

S1: The sum of two of the numbers that turned up was 10
This statement is not sufficient because we cannot determine whether the 4 we already know about is among those two dice

Case a: The two dice that add to 10 are 5 and 5, and the other die is the 4. In this case, the answer to the target question is the sum = 5 + 5 + 4 = 14
Case b: The two dice that add to 10 are 6 and 4, and the other die is a 1. In this case, the answer to the target question with certainty, S1 is not sufficient

S2: The sum of two of the numbers that turned up was 11
This statement doesn’t have the same issue that S1 had.
Since the sum of the two mentioned dice is 11, we can be certain that those two dice are 5, and 6, since those are the only two possible values that will add to 11.
This means the three dice are 5, 6, and 4
So, the answer to the target question is the sum = 5 + 6 + 4 = 15

Since we can answer the target question with certainty, S2 is sufficient

Correct Option: B

“Three dice, each with faces numbered 1 through 6, were tossed onto a game board. If one of the dice turned up 4, what was the sum of the numbers that turned up on all three dice?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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