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

Pontus von Brömssen pontus.von.bromssen at gmail.com
Wed Feb 2 19:50:45 CET 2022

```Dear all,

I made some computations for problems #2 and #4.

For problem #2 the optima for N>=1 are 1, 3, 8, 12, 19, ... . For N=5, one
solution is (the diagrams should be readable in a fixed width font):

5  6  7  8 18
11  .  1  . 10
14  . 19  2 16
.  3  9  4  .
15 12  . 13 17

For problem #4, my interpretation is that only the numbers 1 and 2 can be
placed freely. (In #2, it's 1 through N.) The optima for N>=2 are 3, 5, 8,
10, 13, 15, ... . For N=5, it's one better than the solution given in El
Acertijo:

1  .  2  8  .
9  3  5  .  .
.  .  .  .  6
. 10  .  .  .
.  7  .  .  4

Here is a solution for N=7:

.  8  .  6  .  .  .
. 11  2  .  .  .  .
.  .  .  .  5  . 15
3  .  .  1  .  . 10
14  .  .  .  .  .  .
7  . 12  .  .  .  .
4  .  .  .  9 13  .

Yet another version is A350764, which I submitted a couple of weeks ago
(inspired by the numberphile video). This version is like the original
A337663 but on a finite n X k grid. I also allow any number of initial
ones, mainly in order to have a function of only two variables. (I could
instead have done it on a square n X n grid and let the second argument be
the number of initial ones.) I also submitted A350785, which is this game
played on general graphs.

Best regards,

Pontus

On Tue, Feb 1, 2022 at 9:54 PM Neil Sloane <njasloane at gmail.com> wrote:

> 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
> > 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
> >
> >
>
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>

```