Hard Target - Futility Closet

Hard Target

In a special football game, a team scores 7 points for a touchdown and 3 points for a field goal. What’s the largest mathematically unreachable number of points that a team can score (in an infinitely long game)?

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Eleven. For any integer n greater than that, the possible remainders when n is divided by 3 are 0, 1, and 2. If the remainder is 0, then we can get to n by scoring field goals. If the remainder is 1, then n – 13 has remainder 0 and we can reach n by scoring one touchdown and some number of field goals. And if the remainder is 2, then n – 14 has remainder 0 and we can get to n by scoring two touchdowns and some number of field goals. Put simply, all integers n greater than 11 can be written in the form n = 7a + 3b for some nonnegative integers a and b.

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December 16, 2015December 27, 2015 | Puzzles