Forum Archive :
Cube Handling in Races
Bower's modified Thorp count
> I know that there is something called a "Thorpe Count" involving pip
> count, checkers on the ace point, and open points. But I know I saw a
> much more sophisticated Thorpe Count. It involved crossovers, useless
> gaps, and checkers off, and maybe something else, and purported to
> convert to a winning percentage. Does anyone know the exact formula?
In Hoosier Backgammon Newsletter about 3 or 4 years ago, the virtually
unknown (and definitely unsuccessful) Chuck Bower presented a "Modified
Thorp Count" (Note the CORRECT spelling of Edward Thorp's name--no -E-!).
Here is a reproduction of that method.
NOTE: you absolutely must keep track of the sign in this calculation!
For example, if you subtract two positive numbers giving a negative number,
you MUST keep track of the minus sign.
1) Subtract roller's pip count from non-roller's. Multiply by two.
2) From this, subtract 1/4 of roller's pip count.
3) Subtract number of non-roller's covered home board points from number of
roller's covered home board points. Multiply by 2 and add to result of
4) Compare crossovers. Excess crossovers by non-roller are worth 5 each.
Add to result of step 3. (NOTE: excess crossovers for roller are
subsequently worth -5 each.)
5) A missing home board point is a "useless gap" if:
a) all checkers have been born in (that is, no outfield ckrs remain),
b) at least one higher point is occupied, and
c) a roll of THAT gap's number will not fill a lower gap.
(For example: 1, 3, and 5 points are occupied but the 2,4, and 6 are
empty, the 4-point is a useless gap--4's can't take a checker off and
cannot fill an unoccupied point.)
Add 4 for every useless gap for non-roller on the 3-point or higher.
Subtract 4 for every useless gap of roller's on his/her 3-point or
6) NOTE: I CONSIDER THIS STEP OBSOLETE, BUT INCLUDE IT HERE FOR
Subtract 2 for every checker on any of roller's points in excess of 4.
Add 2 for every checker of non-roller in excess of 4 on a point.
7) Add result of these steps to 74. That is roller's cubeless game winning
chances in percent.
Let's look at an example:
1 2 3 4 5 6
x x x x
x x x x
o o o
o o o o
o o o o o
1 2 3 4 5 6
Assume O is on roll.
1) O leads 40-52. Two times the difference is +24 for roller.
2) 1/4 of 40 is 10. Subtract this from 24 giving 14.
3) O has five points covered to only four by X. Net of one point
covered; multiply by 2 and add to 14, giving 16.
4) O needs 14 crossovers to X's 12. Net two, worth -5 each: 16 - 10 = 6.
5) O has a USELESS GAP on the five point. (A five is wasted by being
forced 6/1--stacking a point already covered.) Subtract 4: 6 - 4 = 2.
7) Add the result to 74, giving O 76% winning chances. (Actual cubeless
chances are 71.7% according to Larry Strommen's BPA.)
Here are some "accouting tricks". You are working out roller's winning
chances. "Good things" for roller are positive (bad are negative). Bad
things for NON-roller are positive (good are negative). So the sign of
each term, by step, goes like this:
1) Race lead--good.
2) Race is long--bad. (Advanced readers please note: this is consistent with
original Thorp Count but not used directly in Kleinman Count.)
3) Points covered--good.
5) Useless gaps on 3-point or higher--bad.
6) OBSOLETE! Stacks are bad.
Note that the example chosen (from a real game) missed by about 4%.
That's not too good. (At least you know I didn't "cherry-pick"!) The
method isn't much better than the "Percentage Thorp Count" that I've
written up in this newsgroup previously. Both pale when compared to the
FULL Kleinman Count (which includes gaps and crossover differences as
well). If you REALLY want to nail these positions, that is the way to
go. I'm working on an article which will compare the various methods.
One final comment. Suppose X were on roll. This method calculates
49% cubeless game winning chances (BPA says 54.9%). In a match where O
is closing in on victory (with a sizable lead), knowing this could be
valuable. Here the original (money play only) Thorp method is virtually
c_ray on FIBS
Cube Handling in Races
- Bower's modified Thorp count (Chuck Bower, July 1997)
- Calculating winning chances (Raccoon, Jan 2007)
- Calculating winning chances (OpenWheel+, Nov 2005)
- Doubling formulas (Michael J. Zehr, Jan 1995)
- Doubling in a long race (Brian Sheppard, Feb 1998)
- EPC example (adambulldog+, Jan 2011)
- EPC example: stack and straggler (neilkaz+, Jan 2009)
- EPC examples: stack and straggler (Carlo Melzi+, Dec 2008)
- Effective pipcount (Douglas Zare, Sept 2003)
- Effective pipcount and type of position (Douglas Zare, Jan 2004)
- Kleinman count (Øystein Johansen+, Feb 2001)
- Kleinman count (André Nicoulin, Sept 1998)
- Kleinman count (Chuck Bower, Mar 1998)
- Lamford's race forumla (Michael Schell, Aug 2001)
- N-roll vs n-roll bearoff (David Rubin+, July 2008)
- N-roll vs n-roll bearoff (Gregg Cattanach, Nov 2002)
- N-roll vs n-roll bearoff (Chuck Bower+, Dec 1997)
- Near end of game (Daniel Murphy, Mar 1997)
- Near end of game (David Montgomery, Feb 1997)
- Near end of game (Ron Karr, Feb 1997)
- One checker model (Kit Woolsey+, Feb 1998)
- Pip count percentage (Jeff Mogath+, Feb 2001)
- Pip-count formulas (Tom Keith+, June 2004)
- Thorp count (Chuck Bower, Jan 1997)
- Thorp count (Simon Woodhead, Sept 1991)
- Thorp count questions (Chuck Bower, Sept 1999)
- Value of a pip (Tom Keith, June 2004)
- Ward's racing formula (Marty Storer, Jan 1992)
- What's your favorite formula? (Timothy Chow+, Aug 2012)