# [seqfan] Stepping Stones: Very nice new versions of the puzzle.

Neil Sloane njasloane at gmail.com
Tue Feb 1 19:50:04 CET 2022

```Dear Seqfans, My old friend Rudolfo Kurchan just wrote to me saying that he
and his friends have been studying some new versions of the Stepping Stones
Problem (A337663, which is also described in the Youtube/Numberphile video
Stones on an Infinite Chessboard,

The new problem #1 is: First place stones labeled 1,2,...,n on an infinite
board. anywhere you want.  Then try to add stones n+1, n+2, ..., M, using
the same rules as in the original problem, and try to maximize M. This is
a(n). Right now they don't have enough terms that are provably correct to
make an OEIS sequence.

Then there are four other problems, also very interesting!

See these four web pages:  They are published in the magazine El Acertijo
Number 5, April 1993 under the name "Bosques de Numeros" (Forests of
Numbers)

https://el-acertijo.blogspot.com/2008/06/el-acertijo-05-pagina-08.html

https://el-acertijo.blogspot.com/2008/06/el-acertijo-05-pagina-09.html

https://el-acertijo.blogspot.com/2008/06/el-acertijo-05-pagina-18.html

https://el-acertijo.blogspot.com/2008/07/el-acertijo-07-pagina-15.html

Jaime Poniachik wrote that problem #1 was invented by Diego Kovacs.

If we start with numbers 1 and 2 we can get up to 10

Starting with numbers 1 to 3 = we can get up to 22 by Daniel Valdano

Starting with numbers 1 to 4 = up to number 30 by Daniel Valdano

What are the best solutions starting with numbers from 1 to N ?

Problem #2: Square field by Rodolfo Kurchan

For an NxN board (not an infinite board) starting with numbers from 1 to N
what are the highest numbers you can reach?

3x3 = 8

4x4 = 12 by Hector San Segundo

[I think the problem starts with an N X N empty board. You try to place the
numbers 1, 2, 3, ..., M that satisfy the same rules as usual, and you want
to maximize M. This is b(n). Obviously M <= N.]

Problem #3: Forest of Rooks: Infinite board.

As in the original problem but now numbers are rooks.

We start with numbers 1 and 2 and we go up to what number ?

[I think the rule now is that we can place k in an empty square if the sum
of the numbers that are a rook's move from that square add to k.]

Problem #4: Forest of transparent Queens in Square fields. N X N board.

As in original square field problem #2 but now numbers are queens.

3x3 = 5

4x4 = 8 Gustavo Piñeiro

5x5 = 9

Problem #5: Forest of opaque Queens in Square fields. N XN board.

In last problem queens were transparent, but now they are opaque

3x3 = 6

4x4 = 9

5x5 = 13 by Daniel Valdano6x6 = 16 by Daniel Valdano

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com

```