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Topic: Matheology § 062
Replies: 181   Last Post: Jul 14, 2012 1:48 PM

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LudovicoVan

Posts: 4,164
From: London
Registered: 2/8/08
Re: Matheology § 062
Posted: Jul 9, 2012 8:35 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

"William Hughes" <wpihughes@gmail.com> wrote in message
news:abd1cfc8-e1de-4504-bd7f-bac8fa5efc7a@h9g2000yqi.googlegroups.com...
<snip>

> Balls only enter the vase at a step
> from the statement of the problem
>
> The only steps are n in N
> from the statement of the problem
>
> At every step n in N, the only balls that
> enter the vase are labeled 1+10(n-1) to 10n
> from the statement of the problem


The statement of the problem talks about an "infinite supply of balls",
which justifies a broader approach in terms of ordinals (and the distinction
balls vs. finite/non-finite labels). Anyway, I am not any good at
arithmetic with ordinals, so here it is over N, i.e. your setting. Of
course, the "structure" we get is not as reach as that given by ordinals and
we can only conclude that the limit set is countably infinite. (My
apologies for any trivial mistakes.)

Let N be the set of balls (w.l.o.g. as each ball is uniquely labeled by a
natural number here).

Let n, m in N.

Let V(n) be the vase at step n, defined by the rules of the game as:

V(0) := { }
V(n) := V(n-1) U { m | 10n-9 <= m <= 10n } \ { m | m = n }

Let V(w) be the limit set (the vase at time 0), defined as: V(w) := U_{n<w}.

Using some arithmetic, this set can be expressed as:

V(w) = N \ { m in N | (m-1) = 0 (mod 10) }

We can now define a bijection between V(w) and N by the functions:

f(m) := m - ceil(m/10)
F(n) := n + ceil(n/9)

QED.

-LV




Date Subject Author
7/7/12
Read Matheology § 062
mueckenh@rz.fh-augsburg.de
7/7/12
Read Re: Matheology § 062
FredJeffries@gmail.com
7/7/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
William Hughes
7/8/12
Read Re: Matheology � 062
Virgil
7/8/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/8/12
Read Re: Matheology � 062
Virgil
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
William Hughes
7/8/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/8/12
Read Re: Matheology � 062
Virgil
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology � 062
Virgil
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology � 062
Virgil
7/8/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology § 062
William Hughes
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
William Hughes
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology § 062
William Hughes
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
William Hughes
7/10/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/10/12
Read Re: Matheology � 062
Virgil
7/10/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/10/12
Read Re: Matheology � 062
Virgil
7/10/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/10/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/10/12
Read Re: Matheology § 062
Gus Gassmann
7/10/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/10/12
Read Re: Matheology � 062
Virgil
7/10/12
Read Re: Matheology § 062
LudovicoVan
7/10/12
Read Re: Matheology § 062
William Hughes
7/10/12
Read Re: Matheology § 062
LudovicoVan
7/10/12
Read Re: Matheology § 062
William Hughes
7/10/12
Read Re: Matheology § 062
Jim Burns
7/10/12
Read Re: Matheology § 062
LudovicoVan
7/10/12
Read Re: Matheology § 062
Gus Gassmann
7/10/12
Read Re: Matheology § 062
LudovicoVan
7/10/12
Read Re: Matheology § 062
Gus Gassmann
7/10/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/10/12
Read Re: Matheology � 062
Virgil
7/10/12
Read Re: Matheology § 062
Gus Gassmann
7/10/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/10/12
Read Re: Matheology � 062
Virgil
7/10/12
Read Re: Matheology § 062
Gus Gassmann
7/11/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
Gus Gassmann
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology § 062
Gus Gassmann
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/11/12
Read Re: Matheology § 062
LudovicoVan
7/11/12
Read Re: Matheology � 062
Virgil
7/12/12
Read Re: Matheology § 062
Gus Gassmann
7/12/12
Read Re: Matheology § 062
LudovicoVan
7/12/12
Read Re: Matheology § 062
William Hughes
7/12/12
Read Re: Matheology § 062
LudovicoVan
7/12/12
Read Re: Matheology � 062
Virgil
7/12/12
Read Re: Matheology � 062
Virgil
7/12/12
Read Re: Matheology § 062
William Hughes
7/12/12
Read Re: Matheology § 062
Gus Gassmann
7/12/12
Read Re: Matheology § 062
LudovicoVan
7/12/12
Read Re: Matheology § 062
Gus Gassmann
7/12/12
Read Re: Matheology § 062
LudovicoVan
7/12/12
Read Re: Matheology � 062
Virgil
7/12/12
Read Re: Matheology � 062
Virgil
7/12/12
Read Re: Matheology § 062
William Hughes
7/12/12
Read Re: Matheology � 062
Virgil
7/12/12
Read Re: Matheology § 062
Gus Gassmann
7/12/12
Read Re: Matheology � 062
Virgil
7/12/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/12/12
Read Re: Matheology � 062
Virgil
7/12/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/12/12
Read Re: Matheology � 062
Virgil
7/13/12
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mueckenh@rz.fh-augsburg.de
7/13/12
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Marshall
7/13/12
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Virgil
7/13/12
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Virgil
7/13/12
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mueckenh@rz.fh-augsburg.de
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Virgil
7/13/12
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Virgil
7/13/12
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LudovicoVan
7/13/12
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7/10/12
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YBM
7/11/12
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7/11/12
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7/10/12
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YBM
7/11/12
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7/11/12
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Virgil
7/10/12
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7/10/12
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Virgil
7/10/12
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7/10/12
Read Re: Matheology § 062
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7/10/12
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Virgil
7/10/12
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7/12/12
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LudovicoVan
7/14/12
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7/14/12
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LudovicoVan
7/14/12
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William Hughes
7/14/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/10/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/10/12
Read Re: Matheology § 062
Ralf Bader
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
William Hughes
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
William Hughes
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
William Hughes
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
William Hughes
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
William Hughes
7/8/12
Read Re: Matheology § 062
William Hughes
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/8/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/9/12
Read Re: Matheology § 062
LudovicoVan
7/9/12
Read Re: Matheology � 062
Virgil
7/8/12
Read Re: Matheology § 062
mueckenh@rz.fh-augsburg.de
7/8/12
Read Re: Matheology � 062
Virgil
7/7/12
Read Re: Matheology � 062
Virgil
7/8/12
Read Re: Matheology § 062
Graham Cooper
7/9/12
Read Re: Matheology § 062
Sam Gusset

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