I have a list of twenty-one (21) two digit numbers; none are multiples of 11, none include the digits 8, 9, or 0, and no two contain the same two digits. (That is, the list doesn't contain both 57 and 75.)

The challenge: Organize all of these numbers into circular list, where no element of the list shares a digit with either of its neighbors, and the neighbors don't share any digits with each other.

(So, for example, 13-42-67-15 is an acceptable segment of the list, but 12-34-15-67 is not, because 12 & 15 share a '1', and they're both neighbors of 34.

If I need to define it, circular list means that the first element and the last element are neighbors.)

## Separation of Numbers

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### Separation of Numbers

Pseudomammal wrote:Biology is funny. Not "ha-ha" funny, "lowest bidder engineering" funny.

### Re: Separation of Numbers

**Spoiler:**

Am I right, or is my code broken?

In the spirit of taking things too far - the 5x5x5x5x5 Rubik's Cube.

- Torn Apart By Dingos
**Posts:**817**Joined:**Thu Aug 03, 2006 2:27 am UTC

### Re: Separation of Numbers

I got the same result.

**Spoiler:**

- Geekthras
- 3) What if it's delicious?
**Posts:**529**Joined:**Wed Oct 03, 2007 4:23 am UTC**Location:**Around Boston, MA

### Re: Separation of Numbers

I'm bored so...

12 34 56 41 32 57 16 23 45 61 72 53 42 67... Later

12 34 56 41 32 57 16 23 45 61 72 53 42 67... Later

Wait. With a SPOON?!

### Re: Separation of Numbers

Geekthras wrote:I'm bored so...

12 34 56 41 32 57 16 23 45 61 72 53 42 67... Later

That one goes wrong at the fourth number.

I'm doing a full exhaustive check by hand, merely because it's something to do instead of actual work. I've covered about 50% of the possibilities so far. Only one full sequence, and it doesn't wrap around.

EDIT: Apparently I was more than 50% through. It's difficult to tell.

**Spoiler:**

All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

### Re: Separation of Numbers

Mouffles is first to answer correctly, but Token's proof is what I was looking for. The result is counterintuitive...well, at least it was unexpected, when I found it myself. That leads me to extend the problem to a list of 42, where each two digits appear twice, e.g. both 67 and 76 are included. Is that enough freedom to find a solution? I honestly don't know yet, but when I get time, I intend to check....

Edit: Obviously, a solution to the first problem would lead quickly to a solution for this one, just by taking the list, making a copy, reversing the digits in each element of the copy, and appending the copy to the original list.

Edit: Obviously, a solution to the first problem would lead quickly to a solution for this one, just by taking the list, making a copy, reversing the digits in each element of the copy, and appending the copy to the original list.

Last edited by EricH on Tue Oct 23, 2007 10:10 pm UTC, edited 1 time in total.

Pseudomammal wrote:Biology is funny. Not "ha-ha" funny, "lowest bidder engineering" funny.

### Re: Separation of Numbers

Wait, I'm confused. I mean, that proof looks valid, but isn't that essentially saying that KG

EDIT: Oops. I forgot about the second condition. That's the problem. Ignore me.

_{7,2}is not Hamiltonian? Because that seems to contradict this theorem. What am I missing?EDIT: Oops. I forgot about the second condition. That's the problem. Ignore me.

### Re: Separation of Numbers

EricH wrote:Mouffles is first to answer correctly, but Token's proof is what I was looking for. The result is counterintuitive...well, at least it was unexpected, when I found it myself. That leads me to extend the problem to a list of 42, where each two digits appear twice, e.g. both 67 and 76 are included. Is that enough freedom to find a solution? I honestly don't know yet, but when I get time, I intend to check....

Edit: Obviously, a solution to the first problem would lead quickly to a solution for this one, just by taking the list, making a copy, reversing the digits in each element of the copy, and appending the copy to the original list.

Hmm... trying the same method on this one would work, but would take significantly longer. There'd be five possible options for the next number instead of two, so the number of sequences to check would be more than squared from that, then a good bit more from the extra length, then a bit more because there's less restriction on how the sequence starts (and by "bit more" I mean "orders of magnitude more"). And I'm lazy. Someone get a computer on it (I'd do it myself, but see previous sentence.)

All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

### Re: Separation of Numbers

EricH wrote:Obviously, a solution to the first problem would lead quickly to a solution for this one, just by taking the list, making a copy, reversing the digits in each element of the copy, and appending the copy to the original list.

**Spoiler:**

In the spirit of taking things too far - the 5x5x5x5x5 Rubik's Cube.

### Re: Separation of Numbers

Oh, nicely done, Mouffles. I guess I won't be looking for that solution myself.

Pseudomammal wrote:Biology is funny. Not "ha-ha" funny, "lowest bidder engineering" funny.

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