Game state after 176:7

From: Stephen Turner (sret1_at_ntlworld.com)
Date: Tue Feb 05 2002 - 03:54:45 PST


We now have six players. Any more want to jump in? New Michael? Anyone who
wants to do so has just over a day left. And James, you still have time to
have another go if you're quick.

--------------------------------------------------------------------------
ROUND 176                             Round start: Wed 2002-01-30 15:32:50

Player                 Style       Valid until
------                 -----       -----------
Ed Murphy              - 0.5       Mon 2002-02-11 22:23:30
Rich Holmes            + 2.0       Fri 2002-02-08 02:42:41
Factitious             + 2.0       Fri 2002-02-08 00:23:06
Jonathan Van Matre     - 2.5       Wed 2002-02-06 18:03:06
Alan Riddell           + 2.0       Wed 2002-02-06 16:50:44
Others                   0         Wed 2002-02-06 15:32:50
James Willson          - 2.0       Tue 2002-02-05 15:32:50

All times are in GMT and base ten.

--------------------------------------------------------------------------
Rule    By whom               When                        Judgement  Style
----    -------               ----                        ---------  -----
176:1   Alan Riddell          Wed 2002-01-30 16:50:44     VALID      + 2.0
176:2   Jonathan Van Matre    Wed 2002-01-30 18:03:06     VALID      - 2.5
176:3   Ed Murphy             Wed 2002-01-30 19:44:52     VALID      - 1.5
176:4   Factitious            Fri 2002-02-01 00:23:06     VALID      + 2.0
176:5   Rich Holmes           Fri 2002-02-01 02:42:41     VALID      + 2.0
176:6   Ed Murphy             Mon 2002-02-04 22:23:30     VALID      + 1.0
177:7   James Willson         Tue 2002-02-05 08:51:33     INVALID    - 2.0

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176:1                                                          VALID  +2.0
Alan Riddell                                           2002-01-30 16:50:44
>>>>>
Future rules shall write all numbers in base 11. e.g.
0,1,2,3,4,5,6,7,8,9,a,10,11,... and therefore for rest of this round, the
round shall be known as "round 150".
<<<<<

Judgement: No problems.

Style: A promising start, setting a direction and yet leaving future rules
plenty of options. Nice and short too -- I like short rules. I give it +2.

--------------------------------------------------------------------------
176:2                                                          VALID  -2.5
Jonathan Van Matre                                     2002-01-30 18:03:06
>>>>>
Define S as a set containing all base 11 numbers.

Define U as a set containing the letter "R".

Define US as the union of set S and set U.

Then,

All your base 11, "R", belong to US.  QED.

Future Fantasy Rules must use the term "QED", or its lengthier expression
"quod erat demonstrandum".
<<<<<

Judgement: Note that eleven is spelled "10". "11" is twelve. However, that
doesn't invalidate this rule.

Style: Sorry, but I dislike this rule. "All your base are belong to us" is
rather a tired joke by now. It can still be funny (you must have a look at
  http://www.randomdrivel.com/media/ayb3.swf
some time) but this is just a feeble -- and ungrammatical -- contrived pun
because the first rule happened to have the word "base" in it.
  In addition, the rule doesn't advance the round much: the restriction is
easy to obey. "QED" is used wrongly (it should follow a proof, not a plain
statement). And marks off for using "11" when you probably meant "10".
  I'm a judge who is not afraid to use the full range of style points. But
I don't quite give this one the minimum because JVM is a newish player and
because I realise that humour is subjective. -2.5.

--------------------------------------------------------------------------
176:3                                                          VALID  -1.5
Ed Murphy                                              2002-01-30 19:44:52
>>>>>
 A +  a       = 15
(A +  a) - a  = 15 - a
 A + (a  - a) = 15 - a
 A            = 15 - a
 A            = 6          QED

To avoid further alphanumeric confusion, future rules shall choose
algebraic symbols from the ISO-6842 alphabet.  This is the same as
the standard English alphabet, except with these symbols omitted:

  O I Z E H S G L B Q A
  o i z e h s g l b q a
<<<<<

Judgement: A tough one to judge but I'm calling it invalid. The problem is
the "ISO-6842". 6842 = 9000 base 10, and ISO 9000 is a well-known standard
about business processes and quality control. Nothing to do with alphabets
at all. I might have let you get away with it if this was one of the story
telling rounds where we build a fantasy world, but here you should perhaps
have chosen an unused number, or not tried to name a standard at all.

Re-judgement: OK, I accept that I made a mistake here. I did consider this
point -- that rules are only obliged to be consistent with previous rules,
not with reality -- and concluded that to some extent at least, rules _do_
have to be consistent with reality. Otherwise whenever a rule was invalid,
you could just say "ah, but the word means the opposite in my world". What
I failed to take into account in this case, however, was that the word was
_explicitly_ redefined in the fantasy world, and that has to be allowed. I
reverse my previous judgement.

Style: Again, we don't seem to be making much progress. In fact, the maths
appears to be the same as in our real world -- why should the symbols +, -
and = have their normal interpretations, for example? Also failing to obey
your own restrictions is always unstylish.

--------------------------------------------------------------------------
176:4                                                          VALID  +2.0
Factitious                                             2002-02-01 00:23:06
>>>>>
    Fantasy math is equipped to deal with problems others consider
insolvable.  For example, dividing by 0 is often considered to be taboo.  We
can get around this by defining the constant j as the inverse of 0, so that 0
times j is equal to 1.  This makes possible new solutions to many difficult
equations, as can be shown by the following proof that there is a value x
satisfying 0x - 4 = 9:

 0x - 4 = 9
 0x = 12
 x = 12j
QED

    Numbers containing j are not considered real, but are part of the set of
fantasy numbers.  Future rules must each contain at least 1 fantasy number.
<<<<<

Judgement: No problems.

Style: Great, we've now got some new fantasy maths introduced -- we'll see
what effect introducing infinities has in future. Also makes a restriction
on future rules. And it uses base 11.

--------------------------------------------------------------------------
176:5                                                          VALID  +2.0
Rich Holmes                                            2002-02-01 02:42:41
>>>>>
A classic "proof" that 0 = 1 goes as follows:

1j = 1j         property of equality
1j-1j = 0       subtract 1j from both sides
(1-1)j = 0      distributivity of subtraction
0j = 0          1-1 = 0
j = 0/0 = 0j    divide both sides by 0
1 = 0   QED     divide both sides by j

The fallacy is of course that fantasy subtraction (and addition) are
not distributive.  Future rule writers would do well to bear this in
mind.
<<<<<

Judgement: Valid. The usual phrase is "multiplication is distributive over
addition" but the meaning is obvious here, so that's not enough to make it
invalid.

Style: A very nice reply to 176:4, working out the consequences of the new
maths introduced there. You're absolutely right, of course, distributivity
must fail when we allow infinities in our system.
  However, although it's very clever I do doubt whether the restriction is
in fact restrictive at all. It may lead to some subtle traps but I suspect
that it won't in fact cause any trouble.

--------------------------------------------------------------------------
176:6                                                          VALID  +1.0
Ed Murphy                                              2002-02-04 22:23:30
>>>>>
Here is another "proof" that 1 = 0:

 j*0     = 1       (definition of j)
(j*0)*0  = 1*0     (multiply both sides by 0)
 j*(0*0) = 0       (associativity of multiplication)
 j*0     = 0       (replace 0*0 with 0)
 1       = 0       (definition of j)     QED

The fallacy is that multiplication is not associative.


The next valid rule must either prove or disprove that

     fc + rc = (f+r)c unless c is a fantasy number


and the next valid rule after that must either prove or disprove that

     (fr)c = f(rc) unless (f = 0 and c is a fantasy number)
                       or (f is a fantasy number and c = 0)
<<<<<

Judgement: I can't see any problems.

Style: It's a bit too similar to the previous rule, and there is no use of
base 11. But it does make the next two rules do some specific maths, which
is thematic, if maybe a bit too prescriptive. I would give it +1.5, but in
fact I'm only going to give it +1 for a reason which I shall reveal later.

--------------------------------------------------------------------------
176:7                                                        INVALID  -2.0
James Willson                                          2002-02-05 08:51:33
>>>>>

Perhaps the most famous fantasy maths problem is the proof that 1 = 0.
It was commonly believed for centuries, but until the groundbreaking work
of Set, all "proofs" that 1 = 0 were unsound.

Set, of course, is the discoverer of k, a fantasy number.
k is defined as 0^0.  Set used this to construct the first
sound proof that 1 = 0.

   k   = k     (all fantasy numbers equal themselves)
   0^0 = 0^0   (definition of k)
   1   = 0^0   (for all n, n^0 = 1)
   1   = 0     (for all n, 0^n = 0)   QED

This has some interesting implications.
Since 1 = k = 0, both 1 and 0 must be fantasy numbers.
Set extended this result to show that all numbers are fantasy numbers.
(This is called Set's Fantastic Completeness Theorem)

We'll use this to prove the following trivially true:

   fc + rc = (f+r)c unless c is a fantasy number

   Since all numbers c are fantasy numbers, the proposition reduces to

      fc + rc = (f+r)c unless true

   and true -> b is true for arbitrary b.  QED

Set made many important contributions to fantasy maths.

All future rules will describe some such contribution.

<<<<<

Judgement: Invalid. If true => b is true for all b then all statements are
true. But there are false statements, e.g., multiplication is associative.

Style: I was glad to find this blunder. The idea that 1 does in fact equal
0 is superficially amusing, but it implies that all numbers -- or at least
all of our numbers -- are equal, which would cause the whole number system
to collapse. (Of course the bigger leap in your proof is not the existence
of 0^0, but the axioms that n^0 = 1 and 0^n = 0 for all n.)
  There are some positive aspects to this rule. First of all, it exploited
the fact that fantasy numbers haven't yet been defined. (This was also the
reason that I knocked off 0.5 points at 176:6 -- this loophole makes those
theorems much easier to prove or disprove.) Secondly it introduces us to a
mathematician. And finally the notion that all numbers are fantasy numbers
is amusing, and doesn't seem to be problematic. Nevertheless, I think that
this would have caused a lot of trouble, and I'm going to give it -2.0.
--------------------------------------------------------------------------


--
Stephen Turner, Cambridge, UK    http://homepage.ntlworld.com/adelie/stephen/
"This is Henman's 8th Wimbledon, and he's only lost 7 matches." BBC, 2/Jul/01

--
Rule Date: 2002-02-05 11:55:02 GMT


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