The problem is to give the legal position which maximises the
total pipcount of both players.
Actually, I thought I had a solution but now I see that it's not so
obviously legal. And maybe the problem has more merit than I
thought. (I wonder if it's in Backgammon Funfair)
On 3/28/2023 8:36 AM, peps...@gmail.com wrote:
The problem is to give the legal position which maximises the
total pipcount of both players.
Actually, I thought I had a solution but now I see that it's not so obviously legal. And maybe the problem has more merit than IBackgammon Funfair has a chapter on "Most checkers on the bar."
thought. (I wonder if it's in Backgammon Funfair)
It makes the following claim: "It is impossible to have more than
fifteen checkers on the bar." This claim sounds plausible, but the
proof in the book strikes me as sketchy. Here's what it says:
"As shown in chapter 12, it is possible to have fifteen White checkers
on the bar. Then a Red checker can come *to* the bar only by being
hit by a White checker entering *from* the bar. Hence the total on
the bar cannot exceed fifteen."
To make this proof more rigorous, one has to show that it doesn't help
to start from some position other than all 15 of one side's checkers
on the bar. But I think this shouldn't be too hard to prove.
By the way, the Funfair book doesn't seem to address your question explicitly.
On 3/28/2023 8:36 AM, peps...@gmail.com wrote:
The problem is to give the legal position which maximises the
total pipcount of both players.
Actually, I thought I had a solution but now I see that it's not so obviously legal. And maybe the problem has more merit than IBackgammon Funfair has a chapter on "Most checkers on the bar."
thought. (I wonder if it's in Backgammon Funfair)
It makes the following claim: "It is impossible to have more than
fifteen checkers on the bar." This claim sounds plausible, but the
proof in the book strikes me as sketchy. Here's what it says:
"As shown in chapter 12, it is possible to have fifteen White checkers
on the bar. Then a Red checker can come *to* the bar only by being
hit by a White checker entering *from* the bar. Hence the total on
the bar cannot exceed fifteen."
To make this proof more rigorous, one has to show that it doesn't help
to start from some position other than all 15 of one side's checkers
on the bar. But I think this shouldn't be too hard to prove.
By the way, the Funfair book doesn't seem to address your question explicitly.
---
Tim Chow
On Tuesday, March 28, 2023 at 2:19:46 PM UTC+1, Timothy Chow wrote:Backgammon Funfair shows that (a) the most checkers that can be on the bar is 15 and (b) the largest pip count that Red can have while still being smaller than White's is 362 and (c) when Red has 362, the most pips that White can have is 372, so that the
On 3/28/2023 8:36 AM, peps...@gmail.com wrote:
The problem is to give the legal position which maximises the
total pipcount of both players.
Actually, I thought I had a solution but now I see that it's not so obviously legal. And maybe the problem has more merit than IBackgammon Funfair has a chapter on "Most checkers on the bar."
thought. (I wonder if it's in Backgammon Funfair)
It makes the following claim: "It is impossible to have more than
fifteen checkers on the bar." This claim sounds plausible, but the
proof in the book strikes me as sketchy. Here's what it says:
"As shown in chapter 12, it is possible to have fifteen White checkers
on the bar. Then a Red checker can come *to* the bar only by being
hit by a White checker entering *from* the bar. Hence the total on
the bar cannot exceed fifteen."
To make this proof more rigorous, one has to show that it doesn't help
to start from some position other than all 15 of one side's checkers
on the bar. But I think this shouldn't be too hard to prove.
By the way, the Funfair book doesn't seem to address your question explicitly.
---Wait a minute. The 15 claim is total nonsense!
Tim Chow
It's clearly possible for you to put all of my 15 checkers on the bar.
After all, backgammon is a competitive game so you're entitled to
be brutal in this context.
Clearly I can enter with doubles and hit four of your blots in the process. This introduces 4 more blots and enters 1 to leave 18 checkers on the bar.
No wonder, the argument can't be made rigorous. The argument might work
in a bg variant where doubles follow the same rules as non-doubles.
Paul
On Tuesday, March 28, 2023 at 6:36:18 PM UTC+1, peps...@gmail.com wrote:the aggregate of pips of 734 is the meximum that can be reached from the starting position.
On Tuesday, March 28, 2023 at 2:19:46 PM UTC+1, Timothy Chow wrote:
On 3/28/2023 8:36 AM, peps...@gmail.com wrote:
The problem is to give the legal position which maximises the
total pipcount of both players.
Actually, I thought I had a solution but now I see that it's not so obviously legal. And maybe the problem has more merit than IBackgammon Funfair has a chapter on "Most checkers on the bar."
thought. (I wonder if it's in Backgammon Funfair)
It makes the following claim: "It is impossible to have more than fifteen checkers on the bar." This claim sounds plausible, but the
proof in the book strikes me as sketchy. Here's what it says:
"As shown in chapter 12, it is possible to have fifteen White checkers on the bar. Then a Red checker can come *to* the bar only by being
hit by a White checker entering *from* the bar. Hence the total on
the bar cannot exceed fifteen."
To make this proof more rigorous, one has to show that it doesn't help to start from some position other than all 15 of one side's checkers
on the bar. But I think this shouldn't be too hard to prove.
By the way, the Funfair book doesn't seem to address your question explicitly.
---Wait a minute. The 15 claim is total nonsense!
Tim Chow
It's clearly possible for you to put all of my 15 checkers on the bar. After all, backgammon is a competitive game so you're entitled to
be brutal in this context.
Clearly I can enter with doubles and hit four of your blots in the process.
This introduces 4 more blots and enters 1 to leave 18 checkers on the bar.
No wonder, the argument can't be made rigorous. The argument might work
in a bg variant where doubles follow the same rules as non-doubles.
PaulBackgammon Funfair shows that (a) the most checkers that can be on the bar is 15 and (b) the largest pip count that Red can have while still being smaller than White's is 362 and (c) when Red has 362, the most pips that White can have is 372, so that
If you have 15 of your checkers on the bar and roll a double you hit either 0 or 1 of your opponent's checkers - not 4.
Backgammon Funfair shows that (a) the most checkers that can be on the bar is 15 and (b) the largest pip count that Red can have while still being smaller than White's is 362 and (c) when Red has 362, the most pips that White can have is 372, so thatthe aggregate of pips of 734 is the meximum that can be reached from the starting position.
Suppose you have your ace point made and I have my ace point made (after
all, copying Tim's backgammon would be unlikely to do me any harm).
You have a stack of three checkers on your 7 point (primality relevant).
I have a stack of two checkers on my 7 point and one on the bar.
Other than this, we each have ten blots on our outer and inner boards.
It seems to me that the great majority of twenty blots can be picked up
by doubles.
On 3/28/2023 4:21 PM, Raymond Kershaw wrote:the aggregate of pips of 734 is the meximum that can be reached from the starting position.
Backgammon Funfair shows that (a) the most checkers that can be on the bar is 15 and (b) the largest pip count that Red can have while still being smaller than White's is 362 and (c) when Red has 362, the most pips that White can have is 372, so that
The final number of 734 might be correct, but it doesn't follow fromTim - White cannot have 385 pips. The maximum a player could have is 375 = 15*25
the other things you say here. For example, maybe the maximum total
occurs when Red has 350 pips and White has 385 pips.
---
Tim Chow
On Wednesday, March 29, 2023 at 1:12:31 AM UTC+1, Timothy Chow wrote:the aggregate of pips of 734 is the meximum that can be reached from the starting position.
On 3/28/2023 4:21 PM, Raymond Kershaw wrote:
Backgammon Funfair shows that (a) the most checkers that can be on the bar is 15 and (b) the largest pip count that Red can have while still being smaller than White's is 362 and (c) when Red has 362, the most pips that White can have is 372, so that
The final number of 734 might be correct, but it doesn't follow fromTim - White cannot have 385 pips. The maximum a player could have is 375 = 15*25
the other things you say here. For example, maybe the maximum total
occurs when Red has 350 pips and White has 385 pips.
---
Tim Chow
On 3/29/2023 5:43 AM, Raymond Kershaw wrote:that the aggregate of pips of 734 is the meximum that can be reached from the starting position.
On Wednesday, March 29, 2023 at 1:12:31 AM UTC+1, Timothy Chow wrote:
On 3/28/2023 4:21 PM, Raymond Kershaw wrote:
Backgammon Funfair shows that (a) the most checkers that can be on the bar is 15 and (b) the largest pip count that Red can have while still being smaller than White's is 362 and (c) when Red has 362, the most pips that White can have is 372, so
If you can show how to get a 360 and 375 position then I would accept that the aggregate pips are one more than 734.Good point. But what about this: Red has 360 pips andThe final number of 734 might be correct, but it doesn't follow fromTim - White cannot have 385 pips. The maximum a player could have is 375 = 15*25
the other things you say here. For example, maybe the maximum total
occurs when Red has 350 pips and White has 385 pips.
---
Tim Chow
White has 375 pips?
Tim Chow
On Wednesday, March 29, 2023 at 2:18:45 PM UTC+1, Timothy Chow wrote:that the aggregate of pips of 734 is the meximum that can be reached from the starting position.
On 3/29/2023 5:43 AM, Raymond Kershaw wrote:
On Wednesday, March 29, 2023 at 1:12:31 AM UTC+1, Timothy Chow wrote:
On 3/28/2023 4:21 PM, Raymond Kershaw wrote:
Backgammon Funfair shows that (a) the most checkers that can be on the bar is 15 and (b) the largest pip count that Red can have while still being smaller than White's is 362 and (c) when Red has 362, the most pips that White can have is 372, so
Good point. But what about this: Red has 360 pips andThe final number of 734 might be correct, but it doesn't follow fromTim - White cannot have 385 pips. The maximum a player could have is 375 = 15*25
the other things you say here. For example, maybe the maximum total
occurs when Red has 350 pips and White has 385 pips.
---
Tim Chow
White has 375 pips?
Tim ChowIf you can show how to get a 360 and 375 position then I would accept that the aggregate pips are one more than 734.
imagining that we can move our checkers around the board and hit
blots when we're on the bar, which of course is illegal.
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