KA-0139Dice and cube face relationships
5 points, difficulty 3 of 3Level 3–4about 105sOn a die, opposite faces add to 7. A die rests on a table with 3 on top. What do the four side faces add up to?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
What do all six faces add to?
2The strategy
Add the numbers 1 to 6 to get the total of the whole die. Then take away the top and the bottom, which you can find from the opposite-faces rule.
3The full solution
All six faces add to 21. The top is 3, so the bottom is 7 - 3 = 4. The sides are 21 - 3 - 4 = 14.
Solution
The reliable way
All six faces together are 1 + 2 + 3 + 4 + 5 + 6 = 21. The top shows 3, so the bottom is 7 - 3 = 4. The four side faces are everything else: 21 - 3 - 4 = 14. Notice that 3 and 4 always add to 7, so the sides come to 21 - 7 = 14 whatever is on top - the number 3 never mattered. The transferable idea: work with the total and subtract, and look for the quantity the question's specific number does not actually affect.
The elegant way
Top and bottom always add to 7, so the four sides always add to 21 - 7 = 14, whichever face is up.
Why this is on the test: The answer is the same whatever the top face shows, and noticing that is worth far more than the arithmetic.