Alex claims it is possible to obtain any score from 0 to 15 by dropping exactly three coins on a board with scores 0, 1, 3, and 5. Which conclusion follows?

Alex claims it is possible to obtain any score from 0 to 15 by dropping exactly three coins on a board with scores 0, 1, 3, and 5. Which conclusion follows?

• A. The conjecture is valid, since all scores are reachable.
• B. The conjecture is invalid, since it is impossible to obtain a score of 12 or 14.
• C. The conjecture is always true.
• D. Not enough information to judge.

Answer: (B)

Think about what three coins can add up to when each coin shows a value from 0, 1, 3, or 5. Break it into cases by how many fives show up. If none of the three coins shows 5, then you’re adding three numbers from {0, 1, 3}. The reachable sums are 0, 1, 2, 3, 4, 5, 6, 7, and 9 (since 3+3+3 is 9, but you can’t reach 8 with these choices). If exactly one coin shows 5, you have 5 plus two numbers from {0, 1, 3}. That yields 5, 6, 7, 8, 9, and 11. If exactly two coins show 5, you have 10 plus one number from {0, 1, 3}. That gives 10, 11, and 13. If all three show 5, you get 15. Combining all possibilities, the totals you can obtain are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, and 15. The scores 12 and 14 cannot be formed. So the claim that you can obtain every score from 0 to 15 is not valid, since 12 and 14 are impossible.

Think about what three coins can add up to when each coin shows a value from 0, 1, 3, or 5. Break it into cases by how many fives show up.

If none of the three coins shows 5, then you’re adding three numbers from {0, 1, 3}. The reachable sums are 0, 1, 2, 3, 4, 5, 6, 7, and 9 (since 3+3+3 is 9, but you can’t reach 8 with these choices).

If exactly one coin shows 5, you have 5 plus two numbers from {0, 1, 3}. That yields 5, 6, 7, 8, 9, and 11.

If exactly two coins show 5, you have 10 plus one number from {0, 1, 3}. That gives 10, 11, and 13.

If all three show 5, you get 15.

Combining all possibilities, the totals you can obtain are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, and 15. The scores 12 and 14 cannot be formed. So the claim that you can obtain every score from 0 to 15 is not valid, since 12 and 14 are impossible.