Six dice with upper faces erased are as shows. The sum of the numbers of dots…
20232025
Six dice with upper faces erased are as shows.
The sum of the numbers of dots on the opposite face is 7. If the dice (I), (II) and (III) have even number of dots on their bottom faces, then what would be the total number of dots on their top faces?
- A.
7
- B.
11
- C.
12
- D.
14
Attempted by 2 students.
Show answer & explanation
Correct answer: B
Concept
On a standard die, dots on any two opposite faces always add up to 7, so the six values 1-6 split into exactly three fixed pairs: (1, 6), (2, 5) and (3, 4). Each of these sketches shows only two faces of a die (front and right); the top and bottom faces are hidden, and together they must be whichever one of the three pairs is NOT already used by the front value or the right value.
Application
Read off the visible front and right faces of dice (I), (II) and (III), identify the pair each belongs to, and the remaining pair gives the top/bottom values:
Die | Front face | Right face | Pairs used | Remaining pair (top, bottom) |
|---|---|---|---|---|
(I) | 3 | 6 | (3,4) and (1,6) | (2, 5) |
(II) | 5 | 4 | (2,5) and (3,4) | (1, 6) |
(III) | 6 | 4 | (1,6) and (3,4) | (2, 5) |
Every one of the three pairs (1,6), (2,5) and (3,4) has one odd member and one even member, so once the bottom face is fixed as the even member, the top face is forced to be the odd member of that same pair:
Die (I): remaining pair is (2, 5); the even value 2 is the bottom, so the top face is 5.
Die (II): remaining pair is (1, 6); the even value 6 is the bottom, so the top face is 1.
Die (III): remaining pair is (2, 5); the even value 2 is the bottom, so the top face is 5.
Total dots on the top faces of dice (I), (II) and (III) = 5 + 1 + 5 = 11.
Cross-check
Since each remaining pair has exactly one odd member, and the bottom is fixed even, every top face here must be odd - so the three top faces are three odd numbers, and their sum must itself be odd. 11 is odd, which is consistent; 12 and 14 could never arise from adding three odd numbers, confirming the total.