Six dice with upper faces erased are as shown. The sum of the numbers of dots…

Concept: On a standard die, dots on opposite faces always sum to 7, so the six faces split into exactly three opposite pairs: {1, 6}, {2, 5}, and {3, 4}. Any two faces that meet along an edge in a drawing (are visibly adjacent, such as the front face and the side face shown here) can never be opposite each other, so they must belong to two different pairs. That leaves only the third, still-unused pair as the possible values for any other face touching that same corner — including a hidden top face. When the two values of that leftover pair differ in parity (one odd, one even), a clue about the top face being odd or even removes the last ambiguity.

Applying the rule to each even-numbered die (II, IV and VI), whose hidden top face is given to carry an even count:

1. Die (II): its visible faces show 5 (front) and 4 (right), which use up the pairs {2, 5} and {3, 4}. The only pair left is {1, 6}; the even value in it is 6, so the hidden top face of (II) = 6.

2. Die (IV): its visible faces show 2 (front) and 4 (right), which use up the pairs {2, 5} and {3, 4}. The only pair left is {1, 6}; the even value in it is 6, so the hidden top face of (IV) = 6.

3. Die (VI): its visible faces show 4 (front) and 5 (right), which use up the pairs {3, 4} and {2, 5}. The only pair left is {1, 6}; the even value in it is 6, so the hidden top face of (VI) = 6.

Cross-check: all three even-numbered dice independently reduce to the same leftover pair {1, 6}, and 6 is the only even value available in that pair each time — so the result is not a coincidence but a direct, repeatable consequence of the opposite-faces-sum-to-7 rule combined with the front/right dot counts actually printed on each die.

Required total = 6 + 6 + 6 = 18.