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

Neil Sloane njasloane at gmail.com
Tue Feb 1 21:54:36 CET 2022

```Correction: for problem 2, I meant to say "Obviously M <= N^2"

On Tue, Feb 1, 2022 at 1:50 PM Neil Sloane <njasloane at gmail.com> wrote:

> 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
>
>

```