Problem 11 · Bound with neighboring pairs · 3 points
Made by AI and not reviewed by a real person.
Seven bowls sit in a row and hold 44 marbles altogether. Each bowl has a positive whole number of marbles, and neighboring bowls hold different amounts. What is the greatest possible number of marbles in the least-filled bowl?
- 3
- 6
- 7
- 4
- 5
Show a hint
- Think about how few marbles two neighboring bowls could hold together if every bowl were fairly full. Then consider several pairs in the row.
- Pair bowls 1–2, 3–4, and 5–6, leaving bowl 7 on its own. Could every bowl hold at least 6 marbles?
Show the answer and solution
Answer: E — 5
If every bowl held at least 6 marbles, each of the three neighboring pairs would hold at least 6 + 7 = 13. With at least 6 in the remaining bowl, that would require 45 marbles, so 6 is impossible. But the row 5, 12, 5, 6, 5, 6, 5 totals 44 and has different amounts in neighboring bowls. Thus the answer is 5. The tempting answer 6 comes from dividing 44 by 7 without allowing for the neighboring bowls.