Six dice with their upper (top) faces erased are shown below. The sum of the…
2024202320242024
Six dice with their upper (top) faces erased are shown below.
The sum of the numbers of dots on opposite faces is 7. If the even-numbered dice have an odd number of dots on their top faces and the odd-numbered dice have an even number of dots on their bottom faces, what is the total number of dots on their top faces?
- A.
12
- B.
14
- C.
16
- D.
18
Attempted by 3 students.
Show answer & explanation
Correct answer: C
On any standard die, the numbers on each pair of opposite faces always add up to 7 (1↔6, 2↔5, 3↔4). This puzzle adds one more rule: for the six dice shown (I–VI), an even-numbered die's hidden top face must carry an odd number of dots, while an odd-numbered die's hidden bottom face must carry an even number of dots.
Each cube here shows two faces -- front and right. The sum-to-7 rule immediately fixes the opposite back and left faces, leaving exactly one complementary pair of numbers unassigned; the parity rule above then decides which of that pair sits on top.
Die | Front | Right | Back (7-Front) | Left (7-Right) | Remaining pair | Parity rule | Top |
|---|---|---|---|---|---|---|---|
I | 3 | 6 | 4 | 1 | {2, 5} | odd die -> bottom even -> bottom = 2 | 5 |
II | 5 | 4 | 2 | 3 | {1, 6} | even die -> top odd | 1 |
III | 6 | 4 | 1 | 3 | {2, 5} | odd die -> bottom even -> bottom = 2 | 5 |
IV | 2 | 4 | 5 | 3 | {1, 6} | even die -> top odd | 1 |
V | 1 | 5 | 6 | 2 | {3, 4} | odd die -> bottom even -> bottom = 4 | 3 |
VI | 4 | 5 | 3 | 2 | {1, 6} | even die -> top odd | 1 |
Check: for every die the six resolved values -- front, back, left, right, top, bottom -- are exactly {1, 2, 3, 4, 5, 6}, each used once, confirming a valid standard die in every case.
Total dots on the six top faces = 5 + 1 + 5 + 1 + 3 + 1 = 16.