答案 Solution (Issue 020)

解答：

如果n是單數的話，這組數字之中超過半數都是單數，因此必定會在某一點發生有兩個單數相連的狀況。由於兩個單數相加一定會是個雙數，所以那些數字之和都不會是質數。

你可能從加分題已經猜到，這條問題其實和孿生質數猜想息息相關：如果 n + 1 和 n + 3 是孿生質數，那麼 (1, n, 3, n – 2, 5, n – 4, …, n – 1, 2) 的組合將會是問題的其中一個解，因為相連數字加起來只會是3、n + 1 或 n + 3。如果孿生質數猜想是正確的話（即是世界上存在著無限多對孿生質數的話），將會有無限多個n可以使這條問題得出解。

Solution:

For an odd n, more than half of the group of numbers involved in the arrangement are odd numbers. Hence two odd numbers must be next to each other in the arrangement. It is well known that two odd numbers add up to an even number, so that sum can never be a prime number.

This problem also has ties to the Twin Prime conjecture, as you might have guessed from the bonus question; if n + 1 and n + 3 are twin primes, then the arrangement (1, n, 3, n – 2, 5, n – 4, …, n – 1, 2) will be a solution to the problem, since the sums of neighboring terms are either 3, n + 1 or n + 3. If the conjecture is true – i.e. there are infinitely many pairs of twin primes – then there will be infinitely many n such that this puzzle has a solution.