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SMITH and JONES: The amazing "FARE" position

Posted By: Nack Ballard
Date: Thursday, 23 February 2017, at 5:49 p.m.

In Response To: SMITH and JONES (Nack Ballard)

------------------------------------------------
Diagram of the OPENING POSITION,
but with White owning a large cube
------------------------------------------------

At the end of the story, I wrote:


Let's recap what we were just told about this amazing game (in progress):

.....(1) Standard money rules were in effect.
.....(2) The checker play and cube action were perfect on both sides.
.....(3) Each player rolled 57 times (counting the opening roll, won by Smith).
.....(4) The cube was turned 64 times (and therefore not turned only 49 times)!
.....(5) The opening position was showing on the board.

Crikey, was I in a madhouse?

Or... Could Smith's and Jones' claims actually be true? If so, HOW?


Well, I suppose both (madhouse and true) could be the case. So far, I've been able to partially verify Smith's and Jones' assertions. With their permission, I contacted General Brown and Admiral White. These gentlemen haven't finished compiling their game records, but they were kind enough to send me a pivotal position that occurred about midway through the game. See the diagram below.






White

score: 0
pip: 166
Unlimited Game
Jacoby Beaver
pip: 166
score: 0

Blue
XGID=aa-a-ADACA--bB--acada-A-AA:1:1:1:00:0:0:3:0:10

The Smith-and-Jones "FARE" position


Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 63.02% (G:32.85% B:5.25%) 62.95% (G:33.09% B:5.49%)
Opponent Winning Chances: 36.98% (G:9.72% B:0.80%) 37.05% (G:9.81% B:0.79%)
Cubeless Equities +0.536 +1.078
Cubeful Equities
No redouble:+0.762 (-0.012)±0.003 (+0.759..+0.765)
Redouble/Take:+0.773±0.006 (+0.768..+0.779)
Redouble/Pass:+1.000 (+0.227)
Best Cube action: Redouble / Take
Rollout details
41472 Games rolled with Variance Reduction.
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 3 days 12 hours 32 minutes


In this position, one player fanned and the other redoubled. It is correct both to redouble and to take, as you can see from their rollout result. (I double-checked by doing a 4-ply rollout, which indicates a redouble by .008. It is only 1844 trials, but that's enough to make clear that the 3-ply result is reasonable, believable.)

Then the player who had redoubled, himself fanned, and got redoubled back. And then it happened again. And again...

I call this a FARE position, which stands for "Fan And Redouble Eternally" (or the "E" can stand for Echo or Exchange, as you like). As long as the fanning continues, the position is correctly re-cubed every roll. In abstract theory, the fan-and-redouble action can go on forever, in which case the cube reaches an infinite level without the players ever moving another checker.

In the case of the Smith-vs-Jones game, the fan-and-redouble loop eventually ended. But not before it had repeated for 63 rolls. Finally, one of them rolled a number other than double 6s and the players started moving the checkers again.

It is not hard to create a FARE-like position. Just spray a bunch of blots around and give each player at least one inside point and a checker on the bar. (It doesn't need to be symmetrical.) However, a proper FARE position is one that can be reached legally from the opening position with perfect play. It's like threading a needle. If there are too few blots (or none on a high inside point), the position won't be strong enough to redouble. If there are too many blots, the position will be impossible to reach.

[This will give you an idea of the sensitivity of FARE criteria: If you nudge each player's 9pt blot back to his 10pt (a position I checked because the pipcount is cutely 167-167, same as the opening position), it is wrong to redouble by .020.]


So, now we are presented with an interesting task. HOW CAN THIS POSITION (repeated below) BE REACHED WITH PERFECT PLAY? Does anyone have any ideas?

Don't worry about finding a path all the way back to the opening position. It is quite enough to retrace a couple/few checker plays. (Can you go back even one?)

Nack





White

score: 0
pip: 166
Unlimited Game
Jacoby Beaver
pip: 166
score: 0

Blue
XGID=aa-a-ADACA--bB--acada-A-AA:1:1:1:00:0:0:3:0:10

The Smith-and-Jones "FARE" position


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