Problem 12 · Double counting · 4 points
Made by AI and not reviewed by a real person.
An alien lines up moonstones numbered 1 through 8, using each number once, and writes the sum on every gap between neighbors. If the seven sums add to 69, which two numbers are at the ends?
- 7 and 8
- 2 and 3
- 3 and 4
- 1 and 3
- 1 and 2
Show a hint
- How many times does an end number appear in the seven sums?
- Start by counting every number twice.
Show the answer and solution
Answer: E — 1 and 2
The numbers 1 through 8 add to 36, so counting each twice gives 72. Each end number appears in only one gap sum instead of two, so the end numbers add to 72 − 69 = 3. The only two different numbers that work are 1 and 2. The tempting choice 1 and 3 would make the seven sums total 72 − 4 = 68.