# Neighbouring numbers summing to a prime on a 3x3

Can you place distinct numbers from 0 to 9 on a 3x3 grid such that every pair of neighbouring (horizontally and vertically) numbers sum to a prime? Can you find multiple solutions? Note that the placed numbers can only be used once and one number will remain unused. The generated primes can be reused.

Good luck!

## 3 Answers

The answer is:

Yes

Explanation:

650
123
498

Where the generated primes in the rows are 11, 5, 3, 5, 13, 17 respectively, and in the columns 7, 5, 7, 11, 3 and 11.

Further:

You can generate multiple solutions from reflecting and rotating this grid. I don't know if you can make any more solutions which are fundamentally different from this, though I haven't tried.

No, it's not possible.
Due to in 3x3 grid, there are total 12 neighboring pairs. And consider to combination in 0~9, we can have minimal pair result $$0+1=1$$ and maximum pair result $$8+9=17$$. However in the first 12 primes are: $$2,3,5,7,11,13,17,23,29,31,37,41$$. Hence it's not possible.

Update:
After questions has been added "The generated primes can be reused." criteria, I've found one:

0 5 6
3 2 1
8 9 4
And you could get another solution by flip or rotate this to get a new one.

• You can reuse the generated primes. – Dmitry Kamenetsky Sep 24 '19 at 5:37
• Well, you just edited after the answer... – Conifers Sep 24 '19 at 5:41
• Sorry about that. I just clarified the question. I didn't downvote you. – Dmitry Kamenetsky Sep 24 '19 at 5:45
• Please form a well-defined question before the post next time. It's not good when someone has answered but the issuer still modify the definition and let the original answer got wrong... – Conifers Sep 24 '19 at 5:52
• OK nevermind. I just think too much. – Conifers Sep 24 '19 at 5:59

By brute force computer search, I've found

there are 4 distinct solutions (ignoring rotations and reflections)

They are

038
529
614

038
749
612

129
438
705

129
658
703


• I wrote out a big long answer with a mathematical way of calculating all the permutations (84), and then saw your answer and realized that 1 is not prime... :( – GentlePurpleRain Sep 24 '19 at 16:52