Art Benjamin sent the announcement below to a bunch of people, and
I thought that some r.g.b. readers might be interested.

---

Dear friends in Math and Backgammon,

This Thursday, March 3, at 4 PM Pacific (7 PM Eastern) I will be
giving a (virtual) presentation on Solving the Race in Backgammon
to the Johns Hopkins University Applied Mathematics Community (organized
by HUSAM, the Hopkins Undergraduate Society for Applied Mathematics).
The talk will be aimed at undergraduate applied math students who do not
necessarily have a background in backgammon, but I will quickly get them
up to speed with concepts like the pip count and the doubling cube.
For those who saw my presentation in Claremont earlier this month, this
talk will be more streamlined to allow more time for the important
backgammon results at the end.

Title: Solving the Race in Backgammon

Abstract: Backgammon is perhaps the oldest game that is still played
today. It is a game that combines luck with skill, where two players
take turns rolling dice and decide how to move their checkers in the
best possible way. It is the ultimate math game, where players who
possess a little bit of mathematical knowledge can have a big advantage
over their opponents.  Players also have the opportunity to double the
stakes of a game using something called the doubling cube, which—when
used optimally—leads to players winning more in the long run. Optimal
use of the doubling cube relies on a player's ability to estimate their
winning chances at any stage of the game.

When played to completion, every game of backgammon eventually becomes
a race, where each player attempts to remove all of their checkers
before their opponent does. The goal of our research is to be able to
determine the optimal doubling cube action for any racing position,
and approximate the game winning chances for both sides. By calculating
the Effective Pip Count for both players and identifying the positions'
Variance Types, we arrive at a reasonably simple method for achieving
this which is demonstrably superior to other popular methods.

Arthur Benjamin is the Smallwood Family Professor of Mathematics at
Harvey Mudd College. In 2020, he won the inaugural American Backgammon
Tour Online (ABTO) with the best overall performance in a series of 17
national tournaments.  He has written several books that present
mathematics in a fun and magical way.  He earned his PhD in Mathematical
Sciences from JHU in 1989.

Zoom Link for talk:

HTTPS://WSE.ZOOM.US/J/95972756601

Cheers,

Art Benjamin

On February 27, 2022 at 12:15:05 PM UTC-7, Tim Chow wrote:

> I thought that some r.g.b. readers might be interested.

> Title: Solving the Race in Backgammon

> Arthur Benjamin is the Smallwood Family Professor of
> Mathematics at Harvey Mudd College. In 2020, he won
> the inaugural American Backgammon Tour Online (ABTO)
> with the best overall performance in a series of 17
> national tournaments. He has written several books that
> present mathematics in a fun and magical way. He earned
> his PhD in Mathematical Sciences from JHU in 1989.

Another mentally sick gambler math PHD trying to make
his bit of fortune or fame...? Or a colleague of Axel who
will make Axel's findings official and get credit for it...?

In backgammon, when the contact is broken, the player
who is ahead by even a single pip will win more after 4
billion games played on starting at that position, regardless
of the "cube skill" bullshit, if the players who take turns at
gaining the advantage keep doubling at 50%+ chance of
winning..!

Is there a logical/mathematical ifs, ands or buts argument
about this?

Didn't Axel already prove this through his experiment here
even about positions prior to losing contact and, in fact,
starting from the opening roll on...?

Mother loving sick scumbag PHD idiots living in their own
little sick gamblers' world (PHD or not, at that)... :((

MK

On 2/28/2022 6:03 AM, MK wrote:
> Didn't Axel already prove this through his experiment here
> even about positions prior to losing contact and, in fact,
> starting from the opening roll on...?

There is a potentially interesting question along these lines
that I don't think has been investigated before, and Axel might
be in a good position to do so. Start with some racing position
where the player on roll is just barely an equity favorite; e.g.,

XGID=--ABCDE------------ddcbb--:0:0:1:00:0:0:0:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| | | O O O O O |
| | | O O O O O |
| | | O O O |
| | | O O |
| | | |
| |BAR| |
| | | X |
| | | X X |
| | | X X X |
| | | X X X X |
| | | X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 70 O: 66 X-O: 0-0
Cube: 1
X on roll, cube action

The player on roll adopts an equilibrium strategy. The opponent
does not necessarily play an equilibrium strategy. Now we ask,
how many times do they have to play out the position before the
player on roll has a 90% chance of coming out ahead? Call this
number N.

I'd expect that if the opponent's goal is to maximize N, she can
do better than adopt an equilibrium strategy, but this would be
interesting to investigate, and I don't think it has been done
before. If beavers and raccoons are allowed then naturally one
would expect that N can be made even larger.

It might make sense to start with simpler racing positions than
the one above, where one can exhaustively explore the space of
strategies and find what the optimal strategy is (meaning the
one that maximizes N).

---
Tim Chow

On Monday, February 28, 2022 at 12:49:12 PM UTC, Tim Chow wrote:
> On 2/28/2022 6:03 AM, MK wrote:
> > Didn't Axel already prove this through his experiment here
> > even about positions prior to losing contact and, in fact,
> > starting from the opening roll on...?
> There is a potentially interesting question along these lines
> that I don't think has been investigated before, and Axel might
> be in a good position to do so. Start with some racing position
> where the player on roll is just barely an equity favorite; e.g.,
>
> XGID=--ABCDE------------ddcbb--:0:0:1:00:0:0:0:0:10
>
> X:Player 1 O:Player 2
> Score is X:0 O:0. Unlimited Game
> +13-14-15-16-17-18------19-20-21-22-23-24-+
> | | | O O O O O |
> | | | O O O O O |
> | | | O O O |
> | | | O O |
> | | | |
> | |BAR| |
> | | | X |
> | | | X X |
> | | | X X X |
> | | | X X X X |
> | | | X X X X X |
> +12-11-10--9--8--7-------6--5--4--3--2--1-+
> Pip count X: 70 O: 66 X-O: 0-0
> Cube: 1
> X on roll, cube action
>
> The player on roll adopts an equilibrium strategy. The opponent
> does not necessarily play an equilibrium strategy. Now we ask,
> how many times do they have to play out the position before the
> player on roll has a 90% chance of coming out ahead? Call this
> number N.
>
> I'd expect that if the opponent's goal is to maximize N, she can
> do better than adopt an equilibrium strategy, but this would be
> interesting to investigate, and I don't think it has been done
> before. If beavers and raccoons are allowed then naturally one
> would expect that N can be made even larger.
>
> It might make sense to start with simpler racing positions than
> the one above, where one can exhaustively explore the space of
> strategies and find what the optimal strategy is (meaning the
> one that maximizes N).

Obviously, you don't maximise N by theoretically optimal play.
The need to maximise N gives a positive (rather than neutral) value to increasing
the variance. So, if a pass is sufficiently marginal, it becomes a take and if a hold
is sufficiently marginal it becomes a cube.

Devising a complete maximise-N strategy might be too hard, just as optimal backgammon
is too hard a problem.

Why do you think this is interesting? I'm not sure anyone else is interested.

Paul

Timothy Chow <tchow12000@yahoo.com> writes:

> Start with some racing position where the player on roll is just
> barely an equity favorite

[...]

> The player on roll adopts an equilibrium strategy. The opponent does
> not necessarily play an equilibrium strategy. Now we ask, how many
> times do they have to play out the position before the player on roll
> has a 90% chance of coming out ahead? Call this number N.
>
> I'd expect that if the opponent's goal is to maximize N, she can do
> better than adopt an equilibrium strategy

Nice objective function: "I concoct complicated non-equilibrium
strategies as a smoke screen to hide that I have no clue about
simpler equilibrium strategies." (-:

My gut feeling says that "obscurity by volatility" is the way to go.
But I am not keen to waste any research time on this, because my
objective function is different, less exotic one.

Best regards

Axel

"peps...@gmail.com" <pepstein5@gmail.com> writes:

> The need to maximise N gives a positive (rather than neutral) value to
> increasing the variance. So, if a pass is sufficiently marginal, it
> becomes a take and if a hold is sufficiently marginal it becomes a
> cube.

Yes. There is room to go "all-out" until you take every non-gin position
and double whenever it is legal. The more extreme, the better for the
volatility. But I believe there is a limit to the amount of stupidity
(in the sense of equity loss) that can be covered up by volatility (in
the sense of maximizing N).

One could take the hypothetical 8-roll ace-point stack position from
Danny Kleinman, and perhaps easily do something analytical with Markov
chains and simplistic cube strategies (beaver > t %, double > d %).
However, there is one key difference: We cannot end up with a Petersburg
paradox, because the number of rolls has a well-known upper bound and so
the cube has as well. It follows that you cannot "hide" forever.

Just my two cents, I am currently reorganzing my IT environment and do
not yet have access again to my former cube "strategy" work.

Best regards

Axel

On 2/28/2022 8:24 AM, peps...@gmail.com wrote:
> Why do you think this is interesting? I'm not sure anyone else is interested.

You're probably right that very few others are interested; otherwise
it would probably have been done already.

Ultimately, I can only say that I'm following my personal instincts
about what constitutes an interesting research direction. Those
instincts have generally served me well in the past. Finding tractable
but nontrivial problems whose solution answers simple questions about
fundamental mathematical objects is part of that. In this case, this
is one of the first questions I would want to understand when trying
to study non-equilibrium strategies.

---
Tim Chow

On Tuesday, March 1, 2022 at 1:36:06 PM UTC, Tim Chow wrote:
> On 2/28/2022 8:24 AM, peps...@gmail.com wrote:
> > Why do you think this is interesting? I'm not sure anyone else is interested.
> You're probably right that very few others are interested; otherwise
> it would probably have been done already.
>
> Ultimately, I can only say that I'm following my personal instincts
> about what constitutes an interesting research direction. Those
> instincts have generally served me well in the past. Finding tractable
> but nontrivial problems whose solution answers simple questions about
> fundamental mathematical objects is part of that. In this case, this
> is one of the first questions I would want to understand when trying
> to study non-equilibrium strategies.

But backgammon concepts could hardly be further from fundamental mathematical objects.
"Mathematical objects?" -- a definite yes.
But "Fundamental"?? --- Surely you're joking, Mr. Chow?

I'll make up a simple game right now called TossCoin to illustrate my point.
All coin tosses are assumed fair, of course.
Each TossCoin game has a positive integer, n, associated with it.
TossCoin is played with a cube like backgammon, which is used the same way as in backgammon.
Each game starts with an initial coin toss which has the sole purpose of determining who starts the game.
Once the starter is determined, players make alternate coin tosses, accumulating scores, scoring 0 for each tails,
and 1 for each head.
The winner is the first person to exceed their opponent's score by n.

Cube strategy is non-trivial here, and changing the objective of the game via your 90% concept makes the cube strategy even harder.
Good mathematics is about taking the simplest possible examples which illustrate your ideas.
That's (partly) why there have been so few contributions to backgammon theory by people whose primary focus is mathematics.
"One Doug Zare"
"There's only one Douglas Zare!"

There are good reasons why we don't have a plethora of Douglas Zares. Conway just used backgammon to waste time as far as I can tell.

Paul

On 3/1/2022 9:22 AM, peps...@gmail.com wrote:
> But backgammon concepts could hardly be further from fundamental mathematical objects.
> "Mathematical objects?" -- a definite yes.
> But "Fundamental"?? --- Surely you're joking, Mr. Chow?
>
> I'll make up a simple game right now called TossCoin to illustrate my point.

I don't mind asking the same questions about an even simpler game
than backgammon. But I don't think that games that people actually
play "could hardly be further from fundamental mathematical objects."
Mathematics, in the end, is a human activity. What we regard as
fundamental is ultimately based on human preferences. Now I do agree
that a game with lots of finicky, ad hoc rules doesn't really merit
being called fundamental, but backgammon races are pretty simple from
a mathematical point of view.

---
Tim Chow

On Wednesday, March 2, 2022 at 1:50:13 AM UTC, Tim Chow wrote:
> On 3/1/2022 9:22 AM, peps...@gmail.com wrote:
> > But backgammon concepts could hardly be further from fundamental mathematical objects.
> > "Mathematical objects?" -- a definite yes.
> > But "Fundamental"?? --- Surely you're joking, Mr. Chow?
> >
> > I'll make up a simple game right now called TossCoin to illustrate my point.
> I don't mind asking the same questions about an even simpler game
> than backgammon. But I don't think that games that people actually
> play "could hardly be further from fundamental mathematical objects."
> Mathematics, in the end, is a human activity. What we regard as
> fundamental is ultimately based on human preferences. Now I do agree
> that a game with lots of finicky, ad hoc rules doesn't really merit
> being called fundamental, but backgammon races are pretty simple from
> a mathematical point of view.

All mathematics is founded on axioms, as far as I've experienced.
But, of course, to say "That's just an axiom" doesn't shut down the discussion.
We need to check that the axioms make sense.
So the important question is "Why do the axioms make sense?"
I think that axioms should be seen to not be arbitrary, and to be recoverable
from basic human experience (assuming that the human has the requisite knowledge
which may be considerable.)
To me there's an enormous difference between saying:
"Let's add the axiom that each dice has exactly 6 outcomes" and saying
"Let's add the axiom that x + y = y + x."

The first appeals to what seem to be clearly chance developments in human culture --
this seems to be going outside the realm of mathematics.
The second makes sense for a huge variety of human experience that seems very far
from arbitrary. Money is somewhat recent in evolutionary terms. But, if Mary collects
lots of blackberries, and Joe collects lots of blackberries, that's the same thing as if Joe
collects lots of blackberries and Mary collects lots of blackberries.

A natural question (to me) anyway would then be "Ah, but how about the fact that
we count out in base ten. We only do that because we have ten fingers. Isn't that somewhat
arbitrary too?"
But this question totally proves (or at least demonstrates) my point. Mathematics of numbers
is (hardly ever) tied to the base of the number. Ten is just a convenient convention, and nothing
would be fundamentally different if we used base 7 (for example) instead.
When problems do consider the base of the number to be highly relevant --
("What are the properties of the sum of the digits?" etc.) then most mathematicians (including me)
don't think of them as "real mathematics".

[It's not totally clear whether I should be called a "mathematician" or not. I don't have a Ph.D. and I've
never published any research. But let's call me that, for the basis of this discussion.]

Paul

On March 1, 2022 at 6:50:13 PM UTC-7, Tim Chow wrote:

> On 3/1/2022 9:22 AM, peps...@gmail.com wrote:

>> But "Fundamental"?? --- Surely you're joking, Mr. Chow?
>> I'll make up a simple game right now called TossCoin

> ..... backgammon races are pretty simple from
> a mathematical point of view.

By the time I got around to replying to yor first response,
many others got posted and, so, I'll try to give a combined
reply to relevant parts of all of them.

First, sometimes I think I could like you enough to call you
Tim instead of Chow. Now that Paul called you Mr. Chow,
I'll definitely call you Tim. :)

I think you all are tough "nuts" to crack but I have a feeling
that you and Axel will be first math PHD's to call the "cube
skill theory" hopefully total bullshit, less so mostly bullshit
or at least partially bullshit which will be sufficient enough
to declare it "debunked".

Even though he doesn't respond to my doing it, it's good to
see that Axel shows up when you guys call his name and
that he hasn't abandoned his previous experiment, better
yet, he may do more similar experiments.

Paul was obviously wrong to speak for everyone that nobody
was interested in your proposal. In fact, he became interested
himself soon after. His first reaction was more due to denial
of the obvious than lack of interest. He then blabbers up a
deluge of murk. But I will make use of his TossCoin argument.

Now let me offer a few arguments of my own. We know that
an average dice roll has 8.166666666666667 pips. Let's say
8.167 here. Let's also say that if your example position is
played out 4 billion times, (or more if you need), every possible
dice roll will occur the same number of times for each player.
Numbers of same exact positions can be safely assumed to
occur the same number of times for each player also since
this is a "solvable" simple race.

Even though no pair of dice numbers can add up to 8.167 (at
the expense of talking/sounding big) I will offer that we can
statistically/mathematically say that each player will need
their pip count divided by 8.167 times to roll to finish the race.

So, in your example, X will need 70/8.167=8.571 rolls and
O will need 66/8.167=8.081 rolls. Rounded up, both will need
9 rolls and since X is ahead, obviously X will win more.

Let's have fun with borrowing Paul's TossCoin coin, and write
on both sides 8.167 so that heads=tails=8.167 (isn't math
wonderful ;)

If you all agree on my above arguments, instead of using dice,
we can have X and O simply toss "Murat's coin" and advance
8.167 pips at each turn. In a cubeless play, clearly X will win
every time.

Now, let's try to illustrate this to ourselves by using 8 instead
of 8.167 so that we can make actual backgammon moves
and let's also introduce the cube at this point.

First, let's say O is the "mutant" and start bearing of pieces
with actual dice numbers adding up to 8, occasionally editing
the position so that the next turn won't hit an empty point, (it's
okay to do this for the sake of the argument here, since we'll
do it for both sides).

We know that in this situation O will never get ahead and thus
never double. X's cube actions will be all No-double/Take until
the 6th roll when X will still be behind with 30 pips against O's
26 pips but now both will only 4 rolls (x8=32 pips) to win. So,
X will double and O will drop. X will win 1 point and every time
it starts.

Then, let's say X is the "mutant" and try playing the same way.
"Mutant" is ahead and thus will double before even the first
roll every time. If O was a "mutant" also, it would drop every
time and lose only 1 point. But since it's not, it will take and
the game will be played out with O never catching up to X,
and losing 2 points every time.

I think this all should be easy enough for all to follow and
understand but sometimes when things look too simple,
there may be something missed. If anyone can show that
I'm missing something, I won't mind being wrong. Otherwise,
I'm pretty sure that when Axel runs an experiment doing
"real math", he will reach my same conclusions.

Repeat after me: the current s-called "cube skill theory"
is bullshit!

MK

MK <murat@compuplus.net> writes:

> Now let me offer a few arguments of my own.

These arguments are FUBAR.

> in your example, X will need 70/8.167=8.571 rolls and O will need
> 66/8.167=8.081 rolls

No. There is wastage involved. But you will get the benefit of the
ignorant, since that point CAN be repaired.

> let's say O is the "mutant" and start bearing of pieces with actual
> dice numbers adding up to 8, occasionally editing the position so that
> the next turn won't hit an empty point, (it's okay to do this for the
> sake of the argument here, since we'll do it for both sides).

This turns it into a one-checker race on a very large/long backgammon
board. That is the "repair" I mentioned above, an old hat in race theory
from decades ago. So far I am still with you.

> We know that in this situation O will never get ahead and thus never
> double.

True, but a mute point, since the outcome is clear anyway if all rolls
are 8 pips and no wastage/miss can occur.

> X's cube actions will be all No-double/Take until the 6th roll

No. X should double immediately and O should pass, because the outcome
exactly 9 rolls later is deterministic. Because of the fixed roll and
the no-miss shuffling (= one checker race) nothing can go wrong. There
are simply no probabilities involved, and that is the core of the cube
skill.

Best regards

Axel

On 3/2/2022 3:32 AM, peps...@gmail.com wrote:
> ("What are the properties of the sum of the digits?" etc.) then most mathematicians (including me)
> don't think of them as "real mathematics".

It's true that this is the majority opinion. I don't entirely
agree. I partially agree, because questions of that sort *tend* not
to lead very far, and if a line of investigation leads to a dead end,
then I agree that it is not terribly interesting. But I don't for
that reason regard them as "not real mathematics."

It's my belief that non-equilibrium strategies in games involving a
doubling cube are a fruitful topic of research. The question I
suggested is a concrete question to focus on. Its answer is admittedly
not extremely interesting for its own sake, but I have faith that
trying to answer it will lead to interesting insights that will suggest
further lines of investigation.

---
Tim Chow

On Wednesday, March 2, 2022 at 1:01:34 PM UTC, Tim Chow wrote:
> On 3/2/2022 3:32 AM, peps...@gmail.com wrote:
> > ("What are the properties of the sum of the digits?" etc.) then most mathematicians (including me)
> > don't think of them as "real mathematics".
> It's true that this is the majority opinion. I don't entirely
> agree. I partially agree, because questions of that sort *tend* not
> to lead very far, and if a line of investigation leads to a dead end,
> then I agree that it is not terribly interesting. But I don't for
> that reason regard them as "not real mathematics."
>
> It's my belief that non-equilibrium strategies in games involving a
> doubling cube are a fruitful topic of research. The question I
> suggested is a concrete question to focus on. Its answer is admittedly
> not extremely interesting for its own sake, but I have faith that
> trying to answer it will lead to interesting insights that will suggest
> further lines of investigation.

But the standard research methodology (at least in mathematics) is to look for the
simplest unsolved problems, rather than jump straight into backgammon.
So we'd look at the simplest possible games for which a doubling cube makes sense
(which is what I tried to do earlier in the thread).
Perhaps you and I differ in opinion, as to how complex backgammon races are.

Whether or not something should be regarded as "real mathematics" might be a minor
semantic point. If your definition is along the lines of "Can be defined and solved using mathematical
methods" then you're correct. But then we would have to regard all sorts of nonsense as real mathematics like:
"Decode your name with letter substitution. For example 'Tim' would be 20 9 13. Concatenate to make
20913. Convert to base 7. Reverse the digits. Then tell me the square root to 193 decimal places of the
base 7 value."

Anyway, my main point is this: If your problem is interesting (and it very well may be), then my advice
is to look at the simplest possible games, which might not include backgammon.

Paul

On Wednesday, March 2, 2022 at 12:33:26 PM UTC, Axel Reichert wrote:
> MK <mu...@compuplus.net> writes:
>
> > Now let me offer a few arguments of my own.
> These arguments are FUBAR.
They are also foobar in my opinion.
He tends to reproduce almost identical arguments whenever the doubling cube is
discussed. So the arguments function similarly to "foo", "bar" when used
software development texts to indicate placeholders.

Paul

On Sunday, February 27, 2022 at 7:15:05 PM UTC, Tim Chow wrote:
> Art Benjamin sent the announcement below to a bunch of people, and
> I thought that some r.g.b. readers might be interested.
>
> ---
>
> Dear friends in Math and Backgammon,
>
> This Thursday, March 3, at 4 PM Pacific (7 PM Eastern) I will be
> giving a (virtual) presentation on Solving the Race in Backgammon
> to the Johns Hopkins University Applied Mathematics Community (organized
> by HUSAM, the Hopkins Undergraduate Society for Applied Mathematics).
> The talk will be aimed at undergraduate applied math students who do not
> necessarily have a background in backgammon, but I will quickly get them
> up to speed with concepts like the pip count and the doubling cube.
> For those who saw my presentation in Claremont earlier this month, this
> talk will be more streamlined to allow more time for the important
> backgammon results at the end.
>
>
> Title: Solving the Race in Backgammon
>
> Abstract: Backgammon is perhaps the oldest game that is still played
> today. It is a game that combines luck with skill, where two players
> take turns rolling dice and decide how to move their checkers in the
> best possible way. It is the ultimate math game, where players who
> possess a little bit of mathematical knowledge can have a big advantage
> over their opponents. Players also have the opportunity to double the
> stakes of a game using something called the doubling cube, which—when
> used optimally—leads to players winning more in the long run. Optimal
> use of the doubling cube relies on a player's ability to estimate their
> winning chances at any stage of the game.
>
> When played to completion, every game of backgammon eventually becomes
> a race, where each player attempts to remove all of their checkers
> before their opponent does. The goal of our research is to be able to
> determine the optimal doubling cube action for any racing position,
> and approximate the game winning chances for both sides. By calculating
> the Effective Pip Count for both players and identifying the positions'
> Variance Types, we arrive at a reasonably simple method for achieving
> this which is demonstrably superior to other popular methods.
>
> Arthur Benjamin is the Smallwood Family Professor of Mathematics at
> Harvey Mudd College. In 2020, he won the inaugural American Backgammon
> Tour Online (ABTO) with the best overall performance in a series of 17
> national tournaments. He has written several books that present
> mathematics in a fun and magical way. He earned his PhD in Mathematical
> Sciences from JHU in 1989.
>
> Zoom Link for talk:
>
> HTTPS://WSE.ZOOM.US/J/95972756601

Thanks for the announcement, Tim.
I just realized that this is midnight for me in the UK,
and I have a 9 to 6 job.
Admittedly, I work remotely so I don't need to get up that
much before 9, but it still seems a bit unwise to attend this live.

Will the event be recorded, do you know?

Thanks,

Paul

On March 2, 2022 at 5:33:26 AM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:
>> in your example, X will need 70/8.167=8.571 rolls and
>> O will need 66/8.167=8.081 rolls

> No. There is wastage involved. But you will get the benefit
> of the ignorant, since that point CAN be repaired.

There's nothing to repair. A few lines below what you quted,
in the same paragraph, I said "..... if your example position is
played out 4 billion times ..... every possible dice roll .....(and)
same exact positions (will) occur the same number of times
for each player".

"Wastage" is a product of a dice roll and a position. Thus,
after 4 billion games wastage will also be the same for both
players. I don't think you couldn't undestand that because
your are too stupid, which would make you "incurable", but
that your brain blocked it out because of your denial, which
makes you "curable" (or "repairable" if you prefer). And don't
you worry none. I will cure you in time...

>> We know that in this situation O will never get ahead and
>> thus never double.

> True, but a mute point, since the outcome is clear anyway
> if all rolls are 8 pips and no wastage/miss can occur.

It's true regardless of wastage. I was just trying illustrate it
through realistic enough backgammon moves for the "non
mathematicians" he who can't understand numbers... ;)

Similar to Paul's argument about counting in differen bases
than 10, my argument would also be true using odd shaped
and mismatched dice with average pips of 5.321 or 3.998
or 1.001 etc. rounded down to 5, 3, 1 (with the last one being
also the average pip of Paul's coin, rounded up :)

>> X's cube actions will be all No-double/Take until the 6th roll

> No. X should double immediately and O should pass,

In his example position, Tim said "The player on roll adopts
an equilibrium strategy" and X is the player on roll. If you paste
the position ID into your favorite bot that implements your
"cube skill theory" and ask for cube action analysis, it will say
No double/Take.

> the outcome exactly 9 rolls later is deterministic. Because of
> the fixed roll and the no-miss shuffling (= one checker race)
> nothing can go wrong. There are simply no probabilities
> involved, and that is the core of the cube skill.

You're mutating...! You're mutating...! :))

Just add "bullshit" before the final period in your paragraph
and people may think your are Murat's sock puppet... ;)

MK

On March 2, 2022 at 9:50:28 AM UTC-7, peps...@gmail.com wrote:

> On Wednesday, March 2, 2022 at 12:33:26 PM UTC, Axel Reichert wrote:

>> MK <mu...@compuplus.net> writes:
>>> Now let me offer a few arguments of my own.

>> These arguments are FUBAR.

> They are also foobar in my opinion.

If I were holding you guys in my hand at a poker game,
I would've said I have a pair of jackasses, errr, a pair of
jacks I mean. I'll call Axel jackass of spades and Paul
jackass of clubs... :)

> He tends to reproduce almost identical arguments
> whenever the doubling cube is discussed.

What's wron with that? Aren't you all doing the very
same thing??

> So the arguments function similarly to "foo", "bar"
> when used software development texts to indicate
> placeholders.

Irrelevant/constipated pun that didn't work for me but
you can instead impress me if you can answer what
comes after "foo, bar, ..."? "Baz" or "fum"??

MK

On 3/2/2022 11:48 AM, peps...@gmail.com wrote:
> But the standard research methodology (at least in mathematics) is to look for the
> simplest unsolved problems, rather than jump straight into backgammon.
> So we'd look at the simplest possible games for which a doubling cube makes sense
> (which is what I tried to do earlier in the thread).
> Perhaps you and I differ in opinion, as to how complex backgammon races are.

Focusing on the simplest unsolved problems is certainly a
sound general principle.

But there is also value in articulating problems that are
slightly more of a stretch, if they have some kind of sex appeal.

For example, the twin prime conjecture does not meet the criterion
of "simplest unsolved problem" but it is appealing.

---
Tim Chow

On 3/2/2022 4:53 PM, peps...@gmail.com wrote:

> Will the event be recorded, do you know?

I don't know.

---
Tim Chow

On Thursday, March 3, 2022 at 5:15:57 AM UTC, Tim Chow wrote:
> On 3/2/2022 4:53 PM, peps...@gmail.com wrote:
>
> > Will the event be recorded, do you know?
> I don't know.

That's a very ambiguous response, Tim.
Do you mean that you don't know if the event will be recorded?
Or do you mean that you don't know whether you know that the event will be recorded?

(J/k of course).

Paul

On Thursday, March 3, 2022 at 5:13:14 AM UTC, Tim Chow wrote:
> On 3/2/2022 11:48 AM, peps...@gmail.com wrote:
> > But the standard research methodology (at least in mathematics) is to look for the
> > simplest unsolved problems, rather than jump straight into backgammon.
> > So we'd look at the simplest possible games for which a doubling cube makes sense
> > (which is what I tried to do earlier in the thread).
> > Perhaps you and I differ in opinion, as to how complex backgammon races are.
> Focusing on the simplest unsolved problems is certainly a
> sound general principle.
>
> But there is also value in articulating problems that are
> slightly more of a stretch, if they have some kind of sex appeal.
>
> For example, the twin prime conjecture does not meet the criterion
> of "simplest unsolved problem" but it is appealing.

But the twin prime conjecture proves my point, I think.
The interested mathematicians don't immediately try to prove the conjecture.
They focus on simpler (previously) unknown questions, related to the conjecture.
Such as "Does there exist any N such that there are an infinite number of prime pairs (p, p + N)?
(Yes, there are.)

Paul

On March 3, 2022 at 1:13:25 AM UTC-7, peps...@gmail.com wrote:

> On Thursday, March 3, 2022 at 5:15:57 AM UTC, Tim Chow wrote:

>> On 3/2/2022 4:53 PM, peps...@gmail.com wrote:

>>> Will the event be recorded, do you know?

>> I don't know.

> That's a very ambiguous response, Tim.
> Do you mean that you don't know if the event will be
> recorded?
> Or do you mean that you don't know whether you
> know that the event will be recorded?

Your initial inverted question-sentence, omitting the
"if/whether" was a single question, with a single
question mark at the end.

Now you restored/inserted the previously omitted "if"
and "whether" to break it into two questions-sentences.

You're a mother-loving pedantic F... just like Tim who
can't know the difference between "big red truck" and
"red big truck"... You two F... should get married...

> (J/k of course).

"Club" is spelled with a "c" not with a "k", Mr. J/c (i.e.
the "jackass of clubs")...

MK

On 3/3/2022 3:17 AM, peps...@gmail.com wrote:

> But the twin prime conjecture proves my point, I think.
> The interested mathematicians don't immediately try to prove the conjecture.
> They focus on simpler (previously) unknown questions, related to the conjecture.
> Such as "Does there exist any N such that there are an infinite number of prime pairs (p, p + N)?
> (Yes, there are.)

Sure. I'm not prohibiting anyone from tackling my proposed
challenge by solving simpler problems first---that would be
absurd, and unenforceable anyway.

But it doesn't follow that it's a mistake to pose the problem
in the first place.

---
Tim Chow

On Thursday, March 3, 2022 at 12:54:02 PM UTC, Tim Chow wrote:
> On 3/3/2022 3:17 AM, peps...@gmail.com wrote:
>
> > But the twin prime conjecture proves my point, I think.
> > The interested mathematicians don't immediately try to prove the conjecture.
> > They focus on simpler (previously) unknown questions, related to the conjecture.
> > Such as "Does there exist any N such that there are an infinite number of prime pairs (p, p + N)?
> > (Yes, there are.)
> Sure. I'm not prohibiting anyone from tackling my proposed
> challenge by solving simpler problems first---that would be
> absurd, and unenforceable anyway.
>
> But it doesn't follow that it's a mistake to pose the problem
> in the first place.

It's not a mistake to pose the problem. It's a mistake to work on the problem.
Don't run before you can walk, and all that.

But that's my opinion.
Feel free to work on it anyway.

Paul