Cal's Currajong: The Curious Conjecture

The second sprouts conjecture of computer mavens Applegate, Jacobson, and Sleator says that Player1 wins opposite day sprouts for zero and one dots, but loses for two, three and four dots, wins for five and six dots, but loses for seven through nine dots, wins for ten and eleven dots, but loses for twelve through fourteen dots. And so on and so on, all the way up, forever and ever, amen. They proved it through nine dots. (Going up.)

It may be true. It may not be true. But suppose it is true. Then we can ask a couple of interesting questions.

First question: where does Player2 get his advantage? Why does he knock off three games for every two that Player1 grabs? We don't see such an advantage in normal sprouts, at least not through, say, twelve dots. Player1 wins precisely half of the normal games through twelve dots, and Applegate and friends plausibly expect that pattern to continue forever. Why should reverse sprouts be different?

I don't know the answer. Maybe there's something about reverse - I should say misere - sprouts that makes traps more important than in normal sprouts. (Arguably, Player2 is in a better position to throw or avoid a trap.) But I don't really see it.

Second question: why the tsunami? Three is the natural wavelength of sprouts. As enshrined in the Hudson two-thirds rule, by the way. Look at it this way. Three dots tend to add two isolani. If X is losing to traps in N dots, the parity change will make him a trap billionaire in N + 3 dots.

So why the big waves?

The only quasi-answer I can think of is colored by my recent love affair with the five dot flipflop. Could it be that the five dot flipflop, the smallest flipflop there is, has an influence almost as pervasive as the well-known influence of the two dot trap, the smallest trap there is?

Who knows? But maybe a game will soon pop up that makes the whole discussion moot anyway.

ANALYSIS (Find a shrink.) 10- 1(11)1[2-4] 1(11)1[2-5] fails to 1(12)11[6], when player2 will get his desired seven isolani.

But here we have a position in which player1's accurate first move has obviated player2's ability to throw or avoid a trap. Any trap will give player1 the five isolani he needs to avoid stepping into it. But without a trap player1 will get what he ultimately wants: six isolani. 5(12)6 Player2 goes the latter route. 1(13)11[5,7] Player1 wins because six isolani are inevitable.

--Cal Hudson, First World Champion of Sprouts

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