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Subject: Game of Life integers
Date: Wed, 7 Feb 2007 02:01:36 +0100
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Thread-Topic: Game of Life integers
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From: "Eric Angelini" <Eric.Angelini at kntv.be>
To: <seqfan at ext.jussieu.fr>
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Hello SeqFans,
=20
If we represent the ten digits like this (in a 5x3 box):
=20
xxx   x   xxx   xxx   x x   xxx   xxx   xxx   xxx   xxx
x x   x     x     x   x x   x     x       x   x x   x x
x x   x   xxx   xxx   xxx   xxx   xxx     x   xxx   xxx
x x   x   x       x     x     x   x x     x   x x     x
xxx   x   xxx   xxx     x   xxx   xxx     x   xxx   xxx
=20
... the integers which disappear in the "Game of Life"
are listed hereafter (two digits, inside an integer, are=20
always separated by one space):
=20
8,10,14,18,20,31,48,50,81,83,87,88,101,118,122,127,144,148,155,161,174,181,=
188,191,199...
Is the sequence finite?
=20
Best,
E.
=20
[Hand-tested with this applet:
http://www.math.com/students/wonders/life/life.html <http://www.math.com/st=
udents/wonders/life/life.html> ]
=20

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<META NAME=3D"Generator" CONTENT=3D"MS Exchange Server version 6.5.7226.0">
<TITLE>RE: Smallest square-free k-gonal number with n prime factors</TITLE>
</HEAD>
<BODY>
<DIV><FONT color=3D#000000 size=3D2>Hello SeqFans,</FONT></DIV>
<DIV><FONT size=3D2></FONT> </DIV>
<DIV><FONT color=3D#000000 size=3D2>If we represent the ten digits like thi=
s (in=20
a 5x3 box):</FONT></DIV>
<DIV><FONT size=3D2></FONT> </DIV>
<DIV><FONT size=3D2>xxx   x  =20
xxx   xxx   x x   xxx  =20
xxx   xxx   xxx   xxx</FONT></DIV>
<DIV><FONT size=3D2>x x   x    =20
x     x   x x  =20
x     x       x  &nb=
sp;x=20
x   x x</FONT></DIV>
<DIV><FONT size=3D2>x x   x  =20
xxx   xxx   xxx   xxx  =20
xxx     x   xxx   xxx</FONT></DIV>
<DIV><FONT size=3D2>x x   x  =20
x       x    =20
x     x   x x   =20
 x   x x     x</FONT></DIV>
<DIV><FONT size=3D2>xxx   x   xxx  =20
xxx     x   xxx   xxx  &n=
bsp;=20
 x   xxx   xxx</FONT></DIV>
<DIV><FONT size=3D2></FONT> </DIV>
<DIV><FONT size=3D2>... the integers which disappear in the "Game of=20
Life"</FONT></DIV>
<DIV><FONT size=3D2>are listed hereafter </FONT><FONT size=3D2>(two digits,=
 inside=20
an integer, are </FONT></DIV>
<DIV><FONT size=3D2>always separated by one </FONT><FONT=20
size=3D2>space):</FONT></DIV>
<DIV><FONT size=3D2></FONT> </DIV>
<DIV><FONT=20
size=3D2>8,10,14,18,20,31,48,50,81,83,87,88,101,118,122,127,144,148,155,161=
,174,181,188,191,199...</FONT></DIV>
<DIV><FONT size=3D2>Is the sequence finite?</FONT></DIV>
<DIV><FONT size=3D2></FONT> </DIV>
<DIV><FONT size=3D2>Best,</FONT></DIV>
<DIV><FONT size=3D2>E.</FONT></DIV>
<DIV><FONT size=3D2></FONT> </DIV>
<DIV><FONT size=3D2>[Hand-tested with this applet:</FONT></DIV>
<DIV><A href=3D"http://www.math.com/students/wonders/life/life.html"><FONT=
=20
size=3D2>http://www.math.com/students/wonders/life/life.html</FONT></A>]</D=
IV>
<DIV> </DIV>

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Eric Angelini asked:

> If we represent the ten digits like this (in a 5x3 box):
...
> ... the integers which disappear in the "Game of Life" are listed hereafter
> (two digits, inside an integer, are always separated by one space):
> 
> 8,10,14,18,20,31,48,50,81,83,87,88,101,118,122,127,144,148,155,161,174,181,
> 188,191,199...
> Is the sequence finite?

No.  For example, numbers of the form  1811...1181  all die in 5 generations.

Just to clarify, we're using a variable width font, with "1" being narrower
than the other digits; e.g. 1811181 is:

But even if we use a fixed-width font, the same numbers (starting with
181181) still die, in 15 gens:

(Of course, I don't think this sequence should be added to OEIS, since
it depends not only on base-10 representations, but also on a specific
way of representing digits as Life patterns.)

Dean Hickerson
dean at math.ucdavis.edu