[seqfan] Re: « King-walking » integers in a square box

[Andrew Weimholt]

On Fri, Jul 23, 2010 at 7:50 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>
> It seems impossible to find such a
> 4x5 box showing all consecutive in-
> tegers from 0 to 'n' with 'n' > 158.
>

I can confirm that it is indeed impossible.

In order to cover all the numbers from 0 to 99,
every pair of digits need to be adjacent to each other, so we need 2 of each.
In order to cover 100 to 159, the multiset of digits adjacent to
either one of the 1's
is {0,0,2,2,3,3,4,4,5,5,6,7,8,9}.
The only placement of 1's that allows for 14 neighbors among them is

- - - - -
- 1 - - -
- - - 1 -
- - - - -

Furthermore, if digit 'X' occurs in a corner, then the other digit 'X'
occurs in the interior.

>From here we can look at various cases of placing X,Y,Z,W in the
corners and interior
and show why each configuration does not work.

Andrew