**King Walking**

[July, 23^{rd},
2010]

Hello SeqFans,

In this 4x5 box
one can read all consecutive integers from 0 to 158 (included):

** 5 8 0 7
3 **

** 9 6 5 1
8 **

** 1 3 2 4
2 **

** 4 0 9 7
6 **

(grid submitted by **James
Dow Allen** on *rec.puzzles*
two days ago)

The rules are:

- an integer is there if its digits can be walked on by a
chess King (one step in 8 directions: 4 straightly, 4 diagonally)

- two identical digits (or more) can follow each other (as if
the King was jumping on the same square).

Example:

- the integers 58073, 13997 and 13887 are visible below,

- the integer 159 is not:

** 5 8 0 7 3 **

** 9 6 5 1 8 **

** 1 3 2 4 2 **

** 4 0 9 7 6 **

It seems
impossible to find such a 4x5 box showing all consecutive integers from 0 to n
with n > 158.

Here is **Giovanni Restas** 158 solution
or the same box published on *rec.puzzles* yesterday):

** 0 3 6 4
2 **

** 1 7 5 1
3 **

** 4 0 2 8
9 **

** 8 9 6 5
7 **

*Question*:

Using the same
rules, what would be the highest reachable integer in the successive square
boxes 1x1, 2x2, 3x3, 4x4, 5x5, ...

This might
constitute a seq S for Neil. [S starts 0, 3, 8, ...]

Best,

Ι.

__________

[**Andrew Weimholt**]:

> The next term is
at least 58

**8 3 4 9**

**2 0 7 2**

**1 6 5 1**

**9 3 4 8**

I looked a little closer at this
one tonight, and I can now also confirm
that 58 is the next term.

__________

[**Dmitri Kamenetsky**]:

Hi Eric,

What an interesting problem,
thank you! This problem seems perfect
for our competitions: http://www.v-sonline.com/index.pl

I will propose
the problem and if it gets accepted then
we will probably
explore all NxN squares with
N from 3 to 32.

__________

... and so it was the contest asking though for N to be kept between
4 and 13. The sequence (including the substring 0, which was not specified on
*SeqFans*) is currently this one
(September 1^{st}, 2010):

S = 1, 4, 9, 59,
369, 1867, 6389, 37138, 137095, 384057, 490158, 1603594, 4039068, ...

To be followed ?

Best,

Ι.

__________