Six dice with their upper faces erased are as shown below. The sum of the…

On a standard die, the two numbers on any pair of opposite faces always add up to 7 (so 1↔6, 2↔5, and 3↔4). This means that once any two adjacent faces of a die are known, the values on the remaining opposite pair are fixed as a SET — an extra condition (such as ‘the top face is even’) is then needed to say which of the two values is on top and which is on the bottom.

1. The odd-numbered dice are (I), (III) and (V) (the 1st, 3rd and 5th dice in the figure).

2. Die (I): the front face shows 3 and the right face shows 6, so the back face is 7 − 3 = 4 and the left face is 7 − 6 = 1. The remaining (top, bottom) pair is {2, 5}; since the top face must be even, the top face is 2.

3. Die (III): the front face shows 6 and the right face shows 4, so the back face is 7 − 6 = 1 and the left face is 7 − 4 = 3. The remaining (top, bottom) pair is {2, 5}; the even value gives a top face of 2.

4. Die (V): the front face shows 1 and the right face shows 5, so the back face is 7 − 1 = 6 and the left face is 7 − 5 = 2. The remaining (top, bottom) pair is {3, 4}; the even value gives a top face of 4.

5. Adding these top faces: 2 + 2 + 4 = 8.

Check: each derived top value pairs correctly with its bottom value to sum to 7 (2 + 5, 2 + 5, 4 + 3), and every chosen top face — 2, 2 and 4 — is indeed even, exactly as the question requires. The even-numbered dice (II), (IV) and (VI) play no part in this total.

The total number of dots on the top faces of dice (I), (III) and (V) is 8.