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New Math

by Douglas Zare
1 April 2006

Douglas Zare

Archaeologists have told me about the old New Math. At some point in the last millennium, mathematicians attempted to introduce layers of abstraction earlier in the curriculum, so that children could distinguish between the number 3 and a set containing 3, should the need arise. This movement is viewed as a failure, but a new New Math may be emerging from a surprising source: Backgammon. Quite a few fundamental results of arithmetic have been overturned by rollouts.

Please study the following carefully. There will be a brief quiz in your local chouette.

3-1 > 4-1



3-1 > 4-1

3-1 is the best opening roll by far and 4-1 may be the worst, but mathematicians used to tell us that 3-1 was less than 4-1, and equal to 4-2. Backgammon players know better.

4+11 > 4+22



Red to play 2-2.

How do you play 2-2 in the opening? When your opponent hasn't accomplished much, unstacking to make the 4 and 11 points is usually better than making the 4 and 22 points. There may be exceptions when your opponent has made a point on offense, but my rollouts show the plays as essentially tied after an opening 6-1, 3-1, 4-2, or 6-4. Another source of exceptions may occur when your opponent has a blot on your 16 point you can threaten by making the 22 point anchor.

While 2 good things are better than 1, 1 good thing is better than 3.



Red to play 4-3.

Here, accomplishing two decent improvements with 24/21 13/9 is better than making the 5 point 9/5 8/5. This is an efficient use of the whole roll.



Red to play 1-1.

When 6/4(2) is a serious candidate, it is usually right. It's a great use of the spares on your 6 point. This is significantly better than the commonly seen blunder that makes 3 tiny improvements, 24/23 8/7(2) 6/5.

6 > 7



Red to play 2-1.

Although 6-primes are stronger than 5-primes, they are also better than 7-primes, which may be viewed as 6-primes with inconveniently located and stacked spares. 9/7 4/3 is much stronger than 11/9 4/3.

Another place this arises is that positions with 7 points are often brittle rather than strong. This is usually called "Too Many Points," although that terminology predates the discovery that 6 > 7.

15 < 11



Red to play 5-1.

Whenever we might leave a shot in the NEBC chouette, Giant 64 member Herb Gurland always counts the shots quickly, considers the other factors carefully, and invariably makes the play with fewest shots anyway, muttering, "It costs a lot to get hit here." Thus, we were initially surprised to see him play 10/4 in positions like these, but the paradox is resolved by the discovery that 15 < 11.

Even though shots are lethal here, the long-term dangers of breaking the 6 point mean leaving 11 shots is more dangerous than leaving 15.

Summary

While duplication is good, quadruplication is asking for it.

A few common ideas can be more memorable by stating them in ways that seem to violate basic arithmetic.

Happy April 1st

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