From: Stephen Turner (analog-author_at_lists.isite.net)
Date: Thu Feb 07 2002 - 01:15:49 PST
Sorry, Factitious, your proposed 150:a arrived 37 minutes after you became
ineligible, so I wasn't able to consider it. All of a sudden, we only have
two players left, and Rich has only about 17.5 hours (base ten!) to submit
another rule.
--------------------------------------------------------------------------
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 + 4.0 (Thu 2002-02-07 00:23:06)
Jonathan Van Matre - 2.5 (Wed 2002-02-06 18:03:06)
Others 0 (Wed 2002-02-06 15:32:50)
Alan Riddell + 2.0 (Tue 2002-02-05 16:50:44)
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
176:7 James Willson Tue 2002-02-05 08:51:33 INVALID - 2.0
176:8 Factitious Wed 2002-02-06 05:37:10 INVALID + 2.0
176:9 Alan Riddell Wed 2002-02-06 15:17:10 INVALID 0.0
--------------------------------------------------------------------------
176:1 VALID +2.0
Alan Riddell Wed 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 Wed 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 Wed 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 Fri 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 Fri 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 Mon 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 Tue 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.
--------------------------------------------------------------------------
176:8 INVALID +2.0
Factitious Wed 2002-02-06 05:37:10
>>>>>
The statement "fc + rc = (f+r)c unless c is a fantasy number" is false
when f = j, r = 1, and c = 0. (Note that 0 is a real number and thus not a
fantasy number.) To show this, I will use my favorite type of proof,
Reductio Ad Absurdum. In other words, I will show that the statement reduces
to a trivial contradiction, thus demonstrating it to be false.
(j+1)*0 = j*0+1*0 (statement under consideration)
(j+1)*0 = 1+1*0 (definition of j)
(j+1)*0 = 1+0 (0 times a real number equals 0)
(j+1)*0 = 1 (0 plus a real number equals that number)
j+1 = 1/0 (divide both sides by 0)
j+1 = j (definition of j)
1 = 0 (subtract j from both sides)
1 is, of course, not equal to zero. This system of math may be
fantastic, but it should not be absurd.
Since a counterexample exists, "fc + rc = (f+r)c unless c is a fantasy
number" is false.
QED.
I like Reductio Ad Absurdum proofs so much that from now on, all valid
odd-numbered rules must contain at least one.
<<<<<
Judgement: I interpret "all numbers in base 11" in 176:1 to imply that all
numbers must be written in digits. But this rule contains the words "zero"
and "one". (cf "must each contain at least 1 fantasy number" in your first
rule).
Style: I like the proof, and the restriction. Also, the rule ties up a few
loose ends. It was only invalid on a technicality, so I still give it +2.
--------------------------------------------------------------------------
176:9 INVALID 0.0
Alan Riddell Wed 2002-02-06 15:17:10
>>>>>
Define the fantasy number m such that,
m+j = j+m = a
0*m = m*0 = m
and for any real number y,
m+y = y+m = m
then,
(m+j)*0 = a*0 = 0
but,
m*0 + j*0 = m + 1 = m
but m is not zero. Therefore addition is not distributive even under
multiplication by a non-fantasy number.
Also, define the fantasy number k such that, for all numbers non zero
numbers y then
|k| < |y|
k > 0
and if y is real,
k*y = y*k = k
k+k = k
k*k = k
y+k = k+y
<<<<<
Judgement: Fails in the same way as 176:8, by using the word "zero". Also,
doesn't use "QED" as demanded by 176:2.
Style: There are some good things about this rule. I still like the notion
of introducing more fantasy numbers, although these ones have really weird
properties! And using the number a is cute.
However, this rule also has the appearance of being written in haste. It
is invalid in careless ways. And although it progresses the game by making
new maths, it doesn't (at least explicitly) constrain future rules. Lastly
you got your rule number wrong in the header of your mail. Weighing it all
up, I balance it out slightly negative, at -1.0.
Style re-assessment: Apparently Alan never received 176:8 or the judgement
on it. Given this, he hadn't been warned about the trap concerning "zero",
and his rule number would have been correct. So I shall re-score this rule
at 0.0 style points.
--------------------------------------------------------------------------
--
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: 2002-02-07 09:16:08 GMT
This archive was generated by hypermail 2.1.5 : Thu Nov 24 2011 - 10:48 PST