Having looked at 9+ recently, and having found that opening to be unusably simple, I have been reminded of an idea I had a long time ago. This is just a speculation, based upon a very small amount of experience, and it will probably not hold up; but, it's an interesting idea.
As we add spots to the starting position, we are adding to the "fecundity" of the position, are increasing the potential of the position to produce survivors. Perhaps it is roughly true that there is a "fecundity factor" f associated with a generalized spot such that n*f has something to say about the total number of survivors to be produced by a starting position. Or n*f might be too simple a formula. But each point added "pushes" the fecundity forward.
Now, the "push" given is less than one. I'm still not back to my best sprouts knowledge, but I think I remember that in normal and misere sprouts seven spots produce five survivors. The 9+ analysis showed nine spots producing six survivors, in normal sprouts. In any event, the forward push is definitely less than one. We do not get another whole survivor every time we add a spot to the starting position.
So the push is like a real number and less than one. But survivors, of course, are measured with integers. We get a kind of interesting conflict. Sometimes the additional push given by an added spot is sufficient to produce an additional survivor, and sometimes not. Actually the formula n*f would lead us to expect regularity in the spot/survivor pattern. The fecundity factor f is insufficient to produce a survivor for every spot, so there should be a regular pattern of "falling behind" and eventual failure, like two flashing lights going in and out of phase, or as if a conveyer belt were going slightly too fast for a worker to smoothly complete some periodic task.
If we look at the little bit we know about sprouts openings, we do notice a (perhaps illusory) pattern. A starting position of one spot robustly produces one survivor. But two spots constitute the famous (in my little, solipsistic world) "universal trap". Two spots produce two survivors in normal sprouts, but one survivor in misere sprouts. There are different ways to describe this situation, but one way is to say that two spots "just barely" produce two survivors. Under this description, we can think of two spots as the crisis before the failure.
And sure enough, there is a failure at three spots. Three spots produce only two survivors. We did not get the additional survivor. Now, the two survivors are very robust for three spots, and this fact might be expected, since after a failure we find ourselves with a great sufficiency for the new, lower standard.
Let's continue our interpretation. Four spots produce three survivors both in normal and misere play, but in my opinion not quite as robustly. And five spots is again unstable. Five spots produce four survivors in normal play, but only three in misere play. Again, we have just barely produced our quota. Our assembly line worker is about to miss an assignment.
Sure enough, six spots produces only four survivors, and these are, in my opinion, robust. Seven spots produce five survivors, but in my opinion with more difficulty. So now, if the pattern continues eight spots should be in crisis.
Sadly, I have not completed my analysis of eight spots. I have not completed 8+, and I have not even looked at 8-. But I find 8+ unusually difficult, and perhaps that unusual difficulty is a sufficient indicator of the crisis we are expecting.
Certainly at least part of the pattern holds for nine spots. If eight spots produces six survivors, then there is a failure at nine spots (as we would expect), since nine spots produce six survivors in normal sprouts (as my recent article "Analyzing 9+" showed.) And these six survivors are produced robustly in the normal game. I have not yet looked at 9-.
And, I have not looked much at anything beyond. I did once post an analysis of 12+ on the Swarthmore forum, and I remember that that analysis was surprisingly easy to do (if correct). That easiness would fit the pattern, since we would expect a crisis at 11 spots and robustness at 12 spots.
I recently posted to the Swarthmore forum an explanation of dubious value for the AJS patttern. The present discussion constitutes an alternative explanation, also of dubious value, for that pattern.
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