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Required fields*

Rev

Required fields*

Prime Number Snake

Place numbers 1 to 100 in the cells of the 10 x 10 board below in such a way that consecutive numbers occupy neighboring cells (either horizontally or vertically). Shaded cells should contain only prime numbers.

Answer*

**Partial solution** *[started before a full solution was posted, so may decide to abandon if I don't get anywhere]*

First, take a look at the empty grid and the snake that must fill it:

[![enter image description here][1]][1]

In the above, I'm using the convention that a thick line is a known border, no line (drawn as pale grey due to Excel) for cells that are connected, and a thin black line where connection is unknown.

There are two distinctive and unique features that are clearly visible, mentioned in the "springboard" from Rand al'Thor, which fix the positions of
>! 1, 2, 93

In addition, we know that
>! odd non-primes (underlined below) must be on the underlined squares, and the cells marked with * can only contain one of the even numbers marked in bold (including 100, which is henceforth marked as 0 to avoid needing one-off special formatting). Remaining bold numbers must be in cells adjacent to at least 2 underlined cells.
>!  
>! 1 and 100 are not in corners, so the corners must be connected to the two adjacent cells.

Marking these "clues" on the image (and highlighting in red the parts of the reference snake whose positions are known), I get an initial layout like this:

>! [![enter image description here][2]][2]

Next, observe that from each of the cells marked with *, the snake MUST take a minimum-distance path to one of the nearest grey cells, giving limits to how "twisty" it can be in those regions.

In particular the region in the bottom right seems to have a limited number of valid possibilities...

>! ... or perhaps less limited than I first thought...
>! manually working through what seemed like a limited set of possibilities led to the following possibilities for the bottom-right corner:
>! [![enter image description here][3]][3]
>! ... originally intended as an exhaustive list, but I'm not 100% on that now.

I ran out of insights about how to deduce the (by now already-known) solution, and I was taking too long, so am now abandoning this answer.

Not sure whether etiquette suggests I should delete it completely, or leave in place in case someone else wants to complete an answer not involving guesswork or computer-based searches.

[1]: https://i.sstatic.net/PN4py.png
  [2]: https://i.sstatic.net/7ZaC8.png
  [3]: https://i.sstatic.net/wrHug.png

Edit Summary*

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$\begingroup$ Seeing how there are already two answers with the full solution, I'd say it's too late to post partials. $\endgroup$
– Bass
Commented Jan 24, 2020 at 11:38
3

$\begingroup$ @bass there weren't when I started writing, and looking at them, I see that neither really "show their working" - will see if I get anywhere with actual deduction rather than exhaustive computer search or informed guesswork and "wiggle it until it fits", and if not can abandon this. $\endgroup$
– Steve
Commented Jan 24, 2020 at 12:10

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