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The Question of Bots Cheating

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Andrew Bokelman

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Oct 3, 1998, 3:00:00 AM10/3/98
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Since there is some concern among some that JellyFish and Snowie cheat
when they roll the dice, would it be possible for someone to develop a third
party roll generator, and do you think that that bot makers would support it
by providing a way to plug it into their program? Or for that matter, is
there a way to already plug in a series of rolls and force one of the bots to
use it.

Gary Wong

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Oct 3, 1998, 3:00:00 AM10/3/98
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I see people advocating this from time to time, and I believe Oasya have
published the specifications on how a third-party DLL can generate the
dice for Snowie, and an indepenendent programmer has done something like
what you suggest for JF (Deja News is your friend :-).

Personally, I can't see the point of it. It is already a simple matter
to satisfy yourself (well, some people are never satisfied; for most people,
it's a simple matter) that [insert name of bot] makes the same move in a
given position regardless of the upcoming dice, and that the same dice are
always generated for a given seed regardless of the position; and therefore
the bot is not cheating. (You can invent other hypotheses about how bots
might possibly cheat, but they get very obscure. They make excellent
plots for X-Files episodes, but poor posts to r.g.b.)

Those that hold the opinion that the bots cheat are either unaware of
the arguments along the above lines, or choose to ignore them. (And ignore
the manual dice option, too.) I doubt that anybody who is not convinced
by a simple, straightforward, rational argument or simple solution would
be convinced by any more complex arguments or solutions.

Heck, I get complaints that Abbott cheats, and it's rated below 1500 most
of the time -- no, I can't figure that one out, either.

Cheers,
Gary.
--
Gary Wong, Department of Computer Science, University of Arizona
ga...@cs.arizona.edu http://www.cs.arizona.edu/~gary/

EdmondT

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Oct 4, 1998, 3:00:00 AM10/4/98
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>It is already a simple matter to satisfy yourself (well, some people are never
satisfied; for most people, it's a simple matter) that [insert name of bot]
makes the same move in a given position regardless of the upcoming dice, and
that the same dice are always generated for a given seed regardless of the
position; and therefore the bot is not cheating. ***

>Those that hold the opinion that the bots cheat are either unaware of the
arguments along the above lines, or choose to ignore them. >

Actually, the dice rolls could be different and annoying to humans without it
actually being cheating.

Consider this analogy. In Bridge, people noticed long ago that many more slam
hands are deal with computer programs than in manual play. The reason: random
dealing of cards does not really occur in manual games of bridge, since people
don't really shuffle the deck enough. If you go to a computer generated hand,
you get a different type of hand dealt. Its actually BETTER, but still is
DIFFERENT.

Isn't this possible with the bots and BG? Clearly, it has been proven to my
satisfaction that they are not "cheating." But is it possible that the
actually random dice still "seem strange" -- or at least vary somewhat -- from
manual throws and thus are odd to people who are used to manual throws?

Thoughts?


Edm...@aol.com

Patti Beadles

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Oct 4, 1998, 3:00:00 AM10/4/98
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A third-party roll generator would most likely just shift the
accusations from the bot author to the third party.

-Patti
--
Patti Beadles |
pat...@netcom.com/pat...@gammon.com |
http://www.gammon.com/ | Try to relax
or just yell, "Hey, Patti!" | and enjoy the crisis

Robert Scibelli

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Oct 5, 1998, 3:00:00 AM10/5/98
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A third-party roll generator will not stop the bots from cheating. I
was playing Jellyfish the other night, and just as I was about to
backgammon him for the fifth straight time I heard a voice from behind
me say "over here". I turned around, but there was no one there. When
I turned back, the board was rearranged so that I was an underdog. Go
figure.


Robert Scibelli
xav...@pipeline.com

Robert-Jan Veldhuizen

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Oct 5, 1998, 3:00:00 AM10/5/98
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On 05-oct-98 16:09:53, Robert Scibelli wrote:

RS> A third-party roll generator will not stop the bots from cheating. I
RS> was playing Jellyfish the other night, and just as I was about to
RS> backgammon him for the fifth straight time I heard a voice from behind
RS> me say "over here". I turned around, but there was no one there. When
RS> I turned back, the board was rearranged so that I was an underdog. Go
RS> figure.

Something similar happened to me on FIBS on yet another night. The
server suddenly asked me "** Are you lonely?". I mean, how does it
now??? Something fishy is going on here and it doens't taste half as
good.

--
Zorba/Robert-Jan


Rodrigo Andrade

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Oct 5, 1998, 3:00:00 AM10/5/98
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While it is totally understandable that some people think that Jellyfish
cheats (read below), I find it very unlikely that we'll ever read a post
entitled "Snowie cheats," or something along these lines, for 2 simple
reasons:

1. Snowie is not available to the masses.

2. It would take somebody really stupid to buy a $300 program and think it
cheats!

RODRIGO

P.S. - I do not think JF cheats! It's just that so many people here say it
cheats that maybe these people are not ready to have their butts kicked!

Gary Wong

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Oct 5, 1998, 3:00:00 AM10/5/98
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Robert-Jan Veldhuizen <veld...@xs4all.nl> writes:
> Something similar happened to me on FIBS on yet another night. The
> server suddenly asked me "** Are you lonely?".

I believe it's "** Are you lonesome tonight?"... my favourite FIBS message :-)
To the bored FIBSters out there: describe how to make FIBS generate the
following:

1) ** Are you lonesome tonight?
2) ** You can't talk if you won't listen.
3) ** You can't say nothing.
4) ** But nobody heard you.
5) ** Please use the 'whois' command.

and for extra credit... how many different ways can you get:

** Permission denied

Gary Wong

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Oct 5, 1998, 3:00:00 AM10/5/98
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edm...@aol.com (EdmondT) writes:
> Actually, the dice rolls could be different and annoying to humans without it
> actually being cheating.
>
> Consider this analogy. In Bridge, people noticed long ago that many more slam
> hands are deal with computer programs than in manual play. The reason: random
> dealing of cards does not really occur in manual games of bridge, since people
> don't really shuffle the deck enough. If you go to a computer generated hand,
> you get a different type of hand dealt. Its actually BETTER, but still is
> DIFFERENT.
>
> Isn't this possible with the bots and BG? Clearly, it has been proven to my
> satisfaction that they are not "cheating." But is it possible that the
> actually random dice still "seem strange" -- or at least vary somewhat -- from
> manual throws and thus are odd to people who are used to manual throws?

Perhaps there will be some variation, but I expect it would be very small.
Assuming the player is honest (ie. not deliberately manipulating the dice),
then there are really only two ways that physical dice rolls can vary from
perfect:

1) slight imperfections in a die causing the distribution to be skewed (eg.
perhaps it rolls slightly more 6s than 1s). With precision dice I would
estimate (I don't have any real figures, sorry) that the distribution
would be within a few parts per million of perfect.

2) some dependence on previous rolls (ie. if you pick your dice up carefully
and don't shake them very well and throw them so they don't tumble much,
then there will be some correlation between this roll and the last one).
Since we're assuming an honest player, I expect shaking precision dice and
throwing them vigourously from a lipped cup would mean this correlation
would be even weaker than effect 1).

I am almost certain that these effects would be so small that it would be
impossible for a human to "notice" any deviation from perfection without
carefully measuring a vast number of trials. My guess is that rolling
dice is considerably more random than dealing cards, given that it's very
easy to roll dice correctly (just give them a good shake!) than it is to
shuffle cards thoroughly (I remember reading that it takes seven riffle
shuffles to mix up an ordered deck of cards, though I didn't see a
justification).

Nathan Kennedy

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Oct 6, 1998, 3:00:00 AM10/6/98
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> My guess is that rolling
> dice is considerably more random than dealing cards, given that it's very
> easy to roll dice correctly (just give them a good shake!) than it is to
> shuffle cards thoroughly (I remember reading that it takes seven riffle
> shuffles to mix up an ordered deck of cards, though I didn't see a
> justification).

Well written. Your conclusion is absolutely true-- a decent roll on dice is for all
practical purposes perfectly random... a riffle-shuffle is for almost all game
purposes random enough as well. But a riffle shuffle is far from truly random.
Well known among card sharks, the Gilbreath principle is at the root of many
statistical anomalies (mentioned in last month's Sci American 50 years of Math
Rec). Take a deck of cards stacked alternating red and black. Riffle-shuffle them
all you want, however you want, all day for that matter, then cut the deck. Pick
the top two cards, you get a red card and a black card. Whoopee, 50% chance of that
happening. Pick the next two cards... Whoa, red and black again... 25% chance...
Then the next two cards-- and so on until the deck is expended with each pair being
black and red... and it SHOULD have been a 1 in 2^26 chance of that happening. Yet
it happens again and again. A far cry from truly random.

Nate


Andrew Bokelman

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Oct 6, 1998, 3:00:00 AM10/6/98
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I know the feeling. Last night I was playing Jellyfish and it starting
singing "Disco Inferno" and blowing cigarette smoke in my face.

Chuck Bower

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Oct 6, 1998, 3:00:00 AM10/6/98
to
In article <19981004074459...@ng112.aol.com>,
EdmondT <edm...@aol.com> wrote:

(snip)


>Consider this analogy. In Bridge, people noticed long ago that many more
>slam hands are deal with computer programs than in manual play. The reason:
>random dealing of cards does not really occur in manual games of bridge,
>since people don't really shuffle the deck enough. If you go to a computer
>generated hand, you get a different type of hand dealt. Its actually BETTER,
>but still is DIFFERENT.
>
>Isn't this possible with the bots and BG? Clearly, it has been proven to my
>satisfaction that they are not "cheating." But is it possible that the
>actually random dice still "seem strange" -- or at least vary somewhat -- from
>manual throws and thus are odd to people who are used to manual throws?

Yes, this is possible. A while back I had the idea of collecting human
rolled dice (e.g. from Hal Heinrich's match database) and analyzing them in
a similar fashion to how FIBS dice have been tested. Then I thought "gee,
that is going to be a LOT of work..." and went on to other things. It would
be a nice undergraduate CS 'thesis' project, IMO. Anyone out there want to
take it on??


Chuck
bo...@bigbang.astro.indiana.edu
c_ray on FIBS

illium

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Oct 6, 1998, 3:00:00 AM10/6/98
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Hi Folks

Going off at a slight tangent here but this is how I would design a
backgammon program to cheat, perhaps because I would want a lot of people
to think it was smart and play lots of money to buy it.:-)

Unlike a chess program I wouldn't want it to win a large fraction of the
time maybe just 50-60% of the time, after all I wouldn't want it to look
too good, and as chess programmers have found to their cost a program that
beats everyone hollow has little appeal to poorer players.

So, let me see, I'm a super-duper backgammon program and I'm starting to
lose rather a lot to my human opponent and I arrive at a position where I'm
starting to lose again, what do I do? Well I could just roll the ramdom
number generator on a few steps, take note of the rolls, reset it, and use
this knowledge to good effect over the next few rolls. However, this might
force me to play unusual moves that might get noticed. Better still, I'll
examine all the 21 possible next rolls, calculate the one that gives me the
biggest increase in equity and simply roll it! How? Well I could just click
the random number generator on a few rolls until I find the one I want or I
could just reseed the random number generator with a pre-determined seed.
If I was a really smart program I could even cover my tracks by always
generating the same series of numbers I did this time when the pre-game
seed is entered again!

Interestingly enough, if I were playing a bot in a tournament and I also
had a copy of the program, I'd be quite happy for it to roll the dice!
here's why.
If the programmer has been sloppy and not choosen a completely new random
seed before starting to play then it might have it's initial seed in the
generator, or be very close to the start of the sequence. After the first
few games, noting down the sequence of rolls, it might be intirely feasable
to syncronise another copy of the program so that it is in step with the
tournament bot, having done this it should be easy to step the RNG forward
and know the coming rolls in advance. Even if the seed is far from it's
original value, with enough computer power at your disposal, locking into
the required sequence is not impossible.

I'm a devious little devil aren't I:-)

Best Regards

William Hill, illium on FIBS.


Patti Beadles

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Oct 6, 1998, 3:00:00 AM10/6/98
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A very rare one: "You are alone here."

-P

Robert-Jan Veldhuizen

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Oct 7, 1998, 3:00:00 AM10/7/98
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On 06-oct-98 02:03:20, Gary Wong wrote:

GW> Robert-Jan Veldhuizen <veld...@xs4all.nl> writes:
>> Something similar happened to me on FIBS on yet another night. The
>> server suddenly asked me "** Are you lonely?".

GW> I believe it's "** Are you lonesome tonight?"... my favourite FIBS
GW> message :-)

Yes you're right, I had forgotten the exact words..guess marvin is an
Elvis fan?

GW> To the bored FIBSters out there: describe how to make FIBS
GW> generate the following:

GW> 1) ** Are you lonesome tonight?
GW> 2) ** You can't talk if you won't listen.
GW> 3) ** You can't say nothing.
GW> 4) ** But nobody heard you.

I know all these..

GW> 5) ** Please use the 'whois' command.

Hm, there's a challenge!

Another funny one is

6) You look great.

GW> and for extra credit... how many different ways can you get:

GW> ** Permission denied

That's easy: roll 66

--
Zorba/Robert-Jan


Robert-Jan Veldhuizen

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Oct 7, 1998, 3:00:00 AM10/7/98
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Oh, and:

7) ** Use a mirror to do that.

;-)
--
Zorba/Robert-Jan


H. Bukkjaer

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Oct 7, 1998, 3:00:00 AM10/7/98
to
I haven't slept for many hours so maybe I'm just completely wrong here
but...

Nathan,
Actually in a perfect random shuffled deck of cards the possibility of the
two top cards being different colors is not 50%... On your first draw it's
50,98% - Then if you got the "match" in your first draw, the possibility
raises on the next draw to 51,02% - then it becomes 51,06% then 51,11%.

Finally when you reach the buttom of the deck (with only 2 cards left - a
red and a black) your possibility of drawing different colors is approx.
100% :-)

So the figure 1:2^26 you mention is IMHO far off.

Best regards
H. Bukkjaer


Nathan Kennedy <bl...@hempseed.com> skrev i artiklen
<3619FFEF...@hempseed.com>...

John Goodwin

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Oct 8, 1998, 3:00:00 AM10/8/98
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On Tue, 06 Oct 1998 19:33:03 +0800, Nathan Kennedy
<bl...@hempseed.com> wrote:

>> My guess is that rolling
>> dice is considerably more random than dealing cards, given that it's very
>> easy to roll dice correctly (just give them a good shake!) than it is to
>> shuffle cards thoroughly (I remember reading that it takes seven riffle
>> shuffles to mix up an ordered deck of cards, though I didn't see a
>> justification).
>

>Well written. Your conclusion is absolutely true-- a decent roll on dice is for all
>practical purposes perfectly random... a riffle-shuffle is for almost all game
>purposes random enough as well. But a riffle shuffle is far from truly random.
>Well known among card sharks, the Gilbreath principle is at the root of many
>statistical anomalies (mentioned in last month's Sci American 50 years of Math
>Rec). Take a deck of cards stacked alternating red and black. Riffle-shuffle them
>all you want, however you want, all day for that matter, then cut the deck. Pick
>the top two cards, you get a red card and a black card. Whoopee, 50% chance of that
>happening. Pick the next two cards... Whoa, red and black again... 25% chance...
>Then the next two cards-- and so on until the deck is expended with each pair being
>black and red... and it SHOULD have been a 1 in 2^26 chance of that happening. Yet
>it happens again and again. A far cry from truly random.

This sounds facinating, so much so that I'd like to try it.

Just one question:

What is a "riffle-shuffle" ?

JG


John Greenwood

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Oct 8, 1998, 3:00:00 AM10/8/98
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"H. Bukkjaer" <bukk...@image.dk> wrote:

>I haven't slept for many hours so maybe I'm just completely wrong here
>but...

>Nathan,
>Actually in a perfect random shuffled deck of cards the possibility of the
>two top cards being different colors is not 50%... On your first draw it's
>50,98% - Then if you got the "match" in your first draw, the possibility
>raises on the next draw to 51,02% - then it becomes 51,06% then 51,11%.

>Finally when you reach the buttom of the deck (with only 2 cards left - a
>red and a black) your possibility of drawing different colors is approx.
>100% :-

But its only going to be one red and one black left if the previous 50
cards were exactly 25 red and 25 black so the odds of that have to be
factored into the equation to get the odds of the last two being
different colours.

I note the smiley at the end of your post. If I've missed the point or
any intended humour I'm sure you will let me know!


>So the figure 1:2^26 you mention is IMHO far off.

>Best regards
>H. Bukkjaer


---

John Greenwood


H. Bukkjaer

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Oct 8, 1998, 3:00:00 AM10/8/98
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Well You're right about the 25-25 card part in my calculation of the 100%
chance. What I ment was that all the possibilities I listed was GIVEN the
previous matches. What you have to do to calculate the possibility of draw
different colored cards all the way through the deck is to multiply all the
single possibilities I listed. The complete calculation would go something
like this:

The top cards color doesn't matter, you just pick it up. Now the 2nd card
must be of the opposite color. Since there are only 51 cards left in the
deck and 26 of them have the opposite color, your possibility of drawing
such a card is 26/51 - NOT 26/52 (50%)...

If you got the 1st round correct, then in the 2nd round the same math
applies. Only this time there will be 25 cards of opposite color out of 49
cards so P=25/49. Now for both things to happen P=(26/51)*(25/49).

Note that the fewer cards in the deck, the farther away from 50% each draws
possibility for a "match" goes. If all draws were matched until you reach
the last 2 cards in the deck you're pretty sure that the remaining two
cards match. Now I wrote _approximately_ 100% therefore the smiley ! (I
know - I need to get out more often).

Now it should be obvious that the possibility given in the original post is
completely wrong since:

1) It states that the possibility of getting each single draw correct is
50% GIVEN that all your previous draws were correct.
2) Actually none of the draws have a possibility of 50% - most obviously
not the last draw !

The correct formula reads:
26!/(51*49*...*3*1) (where 26! is 26 factorial)

If you don't believe the above, then take 4 objects (cards or checkers), 2
of one color and 2 of another color, and put them in a bag or something.
Then say to yourself: Draw two objects - if they are of the same color you
win, if they are of different colors you loose... Many people actually buys
this one - they think that P=50% for either case (it's actually 33%/67%).
Play the game, say 20 times, and you should be convinced...

Well enough boring text without BG relevance...
(of course I would like to hear from anyone who can tell me that the above
isn't true)

Best regards
H. Bukkjaer

> >Actually in a perfect random shuffled deck of cards the possibility of
the
> >two top cards being different colors is not 50%... On your first draw
it's
> >50,98% - Then if you got the "match" in your first draw, the possibility
> >raises on the next draw to 51,02% - then it becomes 51,06% then 51,11%.
>
> >Finally when you reach the buttom of the deck (with only 2 cards left -
a
> >red and a black) your possibility of drawing different colors is approx.

> >100% :-)

alan...@my-dejanews.com

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Oct 8, 1998, 3:00:00 AM10/8/98
to

> Just one question:
>
> What is a "riffle-shuffle" ?
>
> JG
>
>

Actually, the term is "rifle-shuffle". In the June, 1962 issue of Scientific
American, Martin Gardner hypothesized out that a deck of cards could be
thoroughly randomized by being shot off the top of a flagpole with a .22
caliber rifle during a hurricane. For some reason, this hypothesis was not
tested empirically until 1997, (Scientific American, Sept. 1997, Letters to
the Editor) at which time it was observed that the hypothesis was all wet.
There has been some controversy (Id., Nov. 1997, Letters) over whether
Garnder intended the cards to be removed from their box prior to the
experiment. All in all a refreshing reminder that the spirit of science
lives on in these troubled times.

-----------== Posted via Deja News, The Discussion Network ==----------
http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own

Patti Beadles

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Oct 8, 1998, 3:00:00 AM10/8/98
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I believe it has also been shown that the rifle shuffle is nonrandom,
due to the collection problem. i.e. after the shuffle, the deck is
reassembled by a dealer who collects the cards from the field. During
this assembly process, bias is introduced by the collector.

-Patti
(and much swearing, too)
--
Patti Beadles | Not just your average purple-haired
pat...@netcom.com/pat...@gammon.com | degenerate gambling adrenaline
http://www.gammon.com/ | junkie software geek leatherbyke
or just yell, "Hey, Patti!" | nethead biker.

Gary Wong

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Oct 9, 1998, 3:00:00 AM10/9/98
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> Just one question:
>
> What is a "riffle-shuffle" ?

I'm not much of a card player, but the riffle shuffle is what I think
most players would consider a standard shuffle. Hold the deck in your
right hand, face down; flip through the deck with your right thumb
until you get half way; move the bottom half of the deck to your left
hand, both thumbs inward; bend the cards upward in the centre and
slowly ease the thumbs upwards so the cards interleave; then bend the
cards downwards at the edges and slowly ease again and they all "flip"
into one shuffled deck in the middle. The description sounds very
long-winded, but you'd know it if you saw it!

I haven't read the reference Nathan gave, but I believe it concerns
the properties of the "perfect shuffle" (which is what a riffle shuffle
would be if you split the deck EXACTLY in half; and when you were
interleaving the cards you got an EXACT sequence alternating one from
the left, one from the right, etc.) I can't perform a perfect shuffle,
though I've tried! I believe a card shark who'd practiced enough manual
dexterity could manage it. With a perfect shuffle, if you had ten
cards in order 0 1 2 3 4 5 6 7 8 9, after a single shuffle they'd
be 0 5 1 6 2 7 3 8 4 9 (then 0 7 5 3 1 8 6 4 2 9; 0 8 7 6 5 4 3 2 1 9 --
hardly "random", as you can see!)

The shuffle I describe wouldn't give exatcly the properties Nathan
mentions (in particular, any deck that started as RBRBRBRB... would
be permuted to RRBBRRBB... after a single shuffle). Perhaps his
trick depends on a certain multiple of shuffles, or the detail of the
shuffles is slightly different.

In case I ever get in trouble for playing cards at work, I could
point out that the perfect shuffle has important applications in
parallel computing (honest!)

In case I ever get in trouble for posting off-topics articles to
r.g.b., I could steal somebody else's puzzle and tack it on the
end of my post. This one is by Bob Koca (what happened to him,
anyway? I haven't heard from him in months) from the August '96
GAMES magazine:

+13-14-15-16-17-18-------19-20-21-22-23-24-+
| X O X | | X X X X X |
| | | |
| | X | |
| | | |
X | | | |
v| |BAR| |
X | | | |
| | | | OOO
| | | | OOO
| | | O | OOO
| X X X X X | | O | OOO
+12-11-10--9--8--7--------6--5--4--3--2--1-+

O (moving anticlockwise) has borne off 12 chequers. Two X's have been
removed from the game, either from the playing field, the bar, or the
bear-off tray. Where must these two pieces go in order to create the
only possible position that could have resulted from legal moves?

(Answer to be posted in a few days if nobody gets it.)

Chuck Bower

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Oct 9, 1998, 3:00:00 AM10/9/98
to
In article <wtpvc2xk...@brigantine.CS.Arizona.EDU>,
Gary Wong <ga...@cs.arizona.edu> wrote:

(snip)


>This one is by Bob Koca (what happened to him, anyway?

Bob is serving in the Peace Corp in Kenya. Probably no internet
access where he is at. ;)


Chuck
bo...@bigbang.astro.indiana.edu
c_ray in FIBS

Murat Kalinyaprak

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Oct 9, 1998, 3:00:00 AM10/9/98
to

Be careful now. If you make such accusations trashing
a product, some people may request that you provide
strong evidence... :)

Don't feel bad though. What it did to you doesn't seem
as nasty as what other people reported it had done to
them. For instance, I remember some guy had reported
that JF had turned his monitor upside down and spitted
green pea soup like stuff at him (or something along
those)...

To some of us, JF doesn't do anything nasty like these.
In fact, it's being even nice enough to let us beat it
more often that it beats us...

MK

Gary Wong

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Oct 11, 1998, 3:00:00 AM10/11/98
to
Gary Wong <ga...@cs.arizona.edu> writes:
> +13-14-15-16-17-18-------19-20-21-22-23-24-+
> | X O X | | X X X X X |
> | | | |
> | | X | |
> | | | |
> X | | | |
> v| |BAR| |
> X | | | |
> | | | | OOO
> | | | | OOO
> | | | O | OOO
> | X X X X X | | O | OOO
> +12-11-10--9--8--7--------6--5--4--3--2--1-+
>
> O (moving anticlockwise) has borne off 12 chequers. Two X's have been
> removed from the game, either from the playing field, the bar, or the
> bear-off tray. Where must these two pieces go in order to create the
> only possible position that could have resulted from legal moves?

Congratulations to Matt Reklaitis, for correctly deducing the answer to Bob's
puzzle:

Best I came up with is that they must be on the 10 and 11 points,
O just having rolled a 6-1, with no legal 6 to play used the 1 to hit X on 16.

The prize for the second question ("Where is Bob"?) is split between Matt
and Chuck :-)

Hope you enjoyed the puzzle! (It was too hard for me, I had to give up and
peek at the answers.)

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