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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!

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3 Answers 3

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7
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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.

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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.

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    $\begingroup$ You can reuse the generated primes. $\endgroup$
    – Dmitry Kamenetsky
    Commented Sep 24, 2019 at 5:37
  • $\begingroup$ Well, you just edited after the answer... $\endgroup$
    – Conifers
    Commented Sep 24, 2019 at 5:41
  • $\begingroup$ Sorry about that. I just clarified the question. I didn't downvote you. $\endgroup$
    – Dmitry Kamenetsky
    Commented Sep 24, 2019 at 5:45
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    $\begingroup$ Well I thought it was well formed... numbers cannot be reused while I didn't say anything about reusing primes. I added the extra bit just to make sure it's clear. $\endgroup$
    – Dmitry Kamenetsky
    Commented Sep 24, 2019 at 5:57
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    $\begingroup$ OK nevermind. I just think too much. $\endgroup$
    – Conifers
    Commented Sep 24, 2019 at 5:59
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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

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    $\begingroup$ 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... :( $\endgroup$
    – GentlePurpleRain
    Commented Sep 24, 2019 at 16:52

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