games born on day n

Richard Guy rkg at cpsc.ucalgary.ca
Fri Nov 23 18:01:28 CET 2001

```(I was talking about) Normal play partizan games:

Day 0:   0

Day 1:   1  -1  *  (and 0)

Day 2:   Eighteen new ones:  2  -2  1/2 -1/2  1*  -1*
+/-1 up upstar down downstar
*2  1|*  1|0  1|0,*  *|-1
0|-1   0,*|-1
Day 3:   1474 - 22  new ones

See the forthcoming paper of Calistrate, Paulhus
and Wolfe (in More Games of No Chance) for more
insight into the structure of `day n' games.

Note that I misdrew the lattice of day 2 games in
various places:

Handbook of Combinatorics, p.2128

Combinatorial Games (PSAM 43) p.15

Games of No Chance, p.55  (where it's overcautiously
described as a poset)

The games  1|0,*  and  0,*|-1  are clearly more
advantageous to  R  and  L  respectively than
the same games but with one less option (i.e.
1|0  &  1|*  and  0|-1  &  *|-1 )  and should be
moved one level nearer to the middle.
Then it can be drawn using just cubes and squares
and is more visually distributive.  I'm indebted
to Dan Calistrate for pointing this out.       R.

On Thu, 22 Nov 2001, John Conway wrote:

> On Thu, 22 Nov 2001, Richard Guy wrote:
>
> > Dean is quite right, and there should be two sequences.
> >
> > All games are `born again' each day after their birthday.
> >
> > David Wolfe, Dan Calistrate & Marc Paulhus have
> > shown that (all) the games born on day  n  form a
> > distributive lattice.  A paper will appear in
> > More Games of No Chance, the proceedings of the
> > 2000 Workshop at eMiSaRI.
>
>    Since the more interesting sequence is the inclusive
>
> one, I was careful to use the phrase "born by day n"
>                                            ^^
> in ONAG (for the misere games problem).  Just what problem
>
> does the following refer to?   JHC.
>
>
> > Newborn games:  1  3  18  1452  ...
> >
> > Games:          1  4  22  1474  ...
> >
> > The number of levels in the lattice (another
> > recordable sequence?)
> >
> >                 1  3   9    45   2949   ...
> >
> > is one more than twice the (total) number
> > of games born on the previous day.
> >
> > Best,   R.
> >
> > On Wed, 21 Nov 2001, Dean Hickerson wrote:
> >
> > > > Will some fanster put this into the approved
> > > > shape for Neil ?  Thanks!
> > >
> > > I'll do it in a few days if noone does it first.  (I don't have my
> > > references handy at the moment.)
> > >
> > > > The following is the sequence of numbers of games
> > > > born on days 0, 1, 2, 3.
> > >
> > > That should probably be "on or before".  E.g. the 4 games on day 1 include
> > > the one from day 0 and 3 new ones.  Perhaps the sequence (1 3 18 1452) that
> > > counts the new ones on day n should also be included.
> > >
> > > > 1 4 22 1474
> > >
> > > For what it's worth, Robert Li and I also counted 1474 games on day 3 when
> > > we were grad students at Berkeley in 1974.  We were working without a
> > > computer, and made no attempt to count the day 4 games.
> > >
> > > Dean Hickerson
> > > dean at math.ucdavis.edu
> >
> >
>
>

```