[Retros] How many (S)PGs?

Francois Labelle flab at EECS.Berkeley.EDU
Wed Feb 9 03:09:35 EST 2005

Andrew Buchanan wrote:

> It's all about finding questions at the right level of difficulty. How

> hard is the following one to answer?


> Which of the billion-odd at-home positions are (S)PGs?

Here's how far I am toward that goal:

My "7.0 moves or less" search visited 196221 distinct at-home diagrams,
and found 10300 PGs and 9983 SPGs.

My "massacre x=5" search visited 375780 distinct at-home diagrams, and
found 26170 PGs.

Combined, there are 567674 distinct at-home diagrams that were tested,
still far from 2^30.

It might be possible to test a significant portion of the total, but I'm
afraid the missing ones will be extremely hard to find or test. At least
as hard as computing massacres with x=8 or so.

Christoph Fieberg wrote:

> You could extend the question to the ultimate question:


> Which of the maximum 30,3*10^45 chess positions are (S)PGs?

The probability that a random legal diagram is a SPG isn't negligible.
It's safe to say that there exists at least 10^30 (S)PGs, too many to

On the other hand, *one-sided* (S)PGs seem much rarer. I wouldn't be
surprised if there were only about 10^9 of them.


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