King Walking

[July, 23rd, 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 its 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 Restas 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.

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[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.

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[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.

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... 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 1st, 2010):

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

To be followed ?

Best,

Ι.

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