Diamonds and circles on three score cards. Two cards show their totals. Work out the missing number before you peek.
The puzzle
Every diamond is worth the same positive whole number, and every circle is worth the same positive whole number (possibly different from the diamond). The three cards read:
- Card A: 2 diamonds and 1 circle = 12
- Card B: 1 diamond and 3 circles = 11
- Card C: 3 diamonds and 2 circles = ?
What number belongs on card C?
Show answer
Answer: 19.
- Write the cards as equations. Let D be the diamond value and C the circle value: 2D + C = 12, and D + 3C = 11.
- From the second equation, D = 11 − 3C. Plug into the first: 2(11 − 3C) + C = 12 → 22 − 6C + C = 12 → 22 − 5C = 12 → 5C = 10 → C = 2.
- Then D = 11 − 3×2 = 5. A diamond is worth 5 and a circle is worth 2.
- Card C is 3 diamonds and 2 circles: 3×5 + 2×2 = 15 + 4 = 19.
Check the given cards: A is 5 + 5 + 2 = 12, and B is 5 + 2 + 2 + 2 = 11. Only this pair of positive whole-number values fits both cards, so the answer is always 19.