Socratic Tutorial 2

Linda: Sir, could I sit down and ask you a question?

Unidentified Gentleman (U.G.): Are you a Moonie?

Linda: Hardly.

U.G.: Please sit down. Would you like half of this cheesecake?

Linda: No thanks. But should you really be eating that?

U.G.: I think so. I'm carrying hybrid twins.

Linda: My former boyfriend pointed you out to me the other day.

U.G.: He's a chess player?

Linda: No, he plays sprouts.

U.G.: I think I can say with some authority that nobody plays sprouts. There has never been a sprouts game played in recorded history.

Linda: Well, my former boyfriend is wild about sprouts.

U.G.: This sounds very suspicious. Are you and your boyfriend...

Linda: Former.

U.G.: ...mixed up with my club?

Linda: Yes! We follow WGOSA through your...

U.G.: WGOSA? I'm talking about ATARG.

Linda: What's ATARG?

U.G.: Alien Threat Assessment and Response Group. We meet once a month at Ryan's Steakhouse and discuss the writings of David Jacobs. This fall, when the weather gets crisp, we will advance to some exercises. Briefcase swaps, target practice, fields, woods, nightscopes, that sort of thing.

Linda: Really?

U.G.: Have you ever experienced missing time?

Linda: Once. In a topology class.

U.G.: Low probability is much more...colorful...than high probability...

Linda: This is very very interesting, but I have a pressing question about sprouts. Let me show you on this napkin. I'm drawing five spots. Okay this is a 5+ game. Now my understanding is that the correct move here is 1(6)1[2,3] and that a UT is in the offing.

U.G.: That sounds pretty accurate. Didn't a Morphy-Steinitz game go that way?

Linda: I don't really understand UT theory.

U.G.: Well why didn't you say so! Okay, let's put aside your napkin for a moment and take down another napkin. I'm drawing two spots. We're playing normal sprouts.

Linda: That's 2+

U.G.: You're taking notes?

Linda: You wouldn't believe what I have in this notebook.

U.G.: Okay, who wins this game, Left or Right?

Linda: Left moves first and Right moves second, correct?

U.G.: Yes.

Linda: That's easy. The two spot game has two cannibals and yields two survivors. Survivors are cannibals left standing at game's end. But Left needs one survivor. One survivor would prove that Left moved last, since any move changes the parity of the number of cannibals. Left loses and Right wins.

U.G.: Very good. Now, who wins in misere play?

Linda: Another easy one. Left now wants to not move last. She wants to keep an even number of cannibals, signifying that she did not move last, since every move changes the parity of the number of cannibals.

U.G.: Right.

Linda: So, Left wins. Since there will be two survivors.

U.G.: Why will there be two survivors?

Linda: Two spots yield two survivors.

U.G.: Who told you that?

Linda: My former boyfriend. Zero spots yield zero survivors. One spot yields one survivor. Two spots yield two survivors. Then there's slippage...

U.G.: That pattern only applies to normal sprouts. In misere sprouts the two spot game yields one survivor. You really need to analyze some of these positions for yourself, by the way.

Linda: In the 2- game Left needs an even number of survivors. But you're saying there will be an odd number of survivors. So Right wins again!

U.G.: Of course. That's why it's a trap. A trap is a configuration which if played as a game is won by the person moving second, both in normal and misere play. The person moving second can control the parity of the number of survivors.

Linda: So a UT is a trap.

U.G.: It's a special trap. It's so simple it pops up everywhere. Sprouts positions are continually evolving in the direction of UTs.

Linda: As they simplify.

U.G.: Yes.

Linda: Okay. Now, if a position contains an actual UT, what is the significance of that?

U.G.: In the majority of cases, the person who is forced to move to a UT loses, because his opponent will then be able to adjust the number of survivors as he wishes.

Linda: If I move to a UT, my opponent can then force the UT to yield an even or odd number of survivors, just as she wishes, and that action will affect the parity of the number of survivors of the entire position.

U.G.: That's the idea.

Linda: So how do I avoid moving to a UT?

U.G.: You already have the tools for doing that. You already know about cannibals and survivors. Let's go back to your original napkin. After 5+ 1(6)1[2,3] let's have Right play 1(7)2. What should Left play now?

Linda: That's easy. Left should play 4(8)6 creating a symmetrical position.

U.G.: I forgot about that possibility. Very good. Good trap crashing. Okay, but here is a new problem. After 5+ 1(6)1[2,3] 1(7)2, move to the sphere S(2) and win.

Linda: I don't know what you mean by "trap crashing".

U.G.: I'll explain shortly, but please work the problem.

Linda: I have to move to the sphere S(2)?

U.G.: Yes. This is an artificial problem but the principle is important. How can Left win?

Linda: I have no idea! The position seems much too complicated to deal with.

U.G.: It's not complicated at all. You've forgotten the UT motif.

Linda: Okay. I take it that the sphere S(4) is a UT now?

U.G.: Absolutely correct. The sphere S(4) is a type 2 UT.

Linda: Type 2?

U.G.: Because of spot # 6. Without spot # 6, the sphere S(4) would be a type 1 UT.

Linda: What's the difference?

U.G.: There is no difference. You avoid moving to either. For instance, to solve the problem I've given you, you need to find a move that will prevent you from ultimately having to move first to the sphere S(4).

Linda: In other words, you have given me a problem, and to solve this problem I need to find a move which will guarantee that I will not be forced to move to the UT.

U.G.: Right.

Linda: I don't know how to find such a move.

U.G.: Use the cannibal/survivor technique. You want to be the last person to move to the sphere S(2). How many survivors do you need for that to happen?

Linda: I'm not sure...

U.G.: How many cannibals are there?

Linda: In the sphere S(2)?

U.G.: Yes.

Linda: One, two, three. No wait. Is spot # 6 in sphere S(2)?

U.G.: Certainly.

Linda: Then three. I thought you said spot # 6 is in sphere S(4).

U.G.: Spot # 6 is a pivot. It resides both in sphere S(2) and sphere S(4).

Linda: Okay. So the sphere S(2) contains three cannibals. That means I want an even number of survivors, right?

U.G.: Right.

Linda: In other words, two survivors.

U.G.: Yes.

Linda: Therefore the move 2(8)2[3] wins.

U.G.: Very good. You've got it.

Linda: The move 6(8)7[2] also wins.

U.G.: Correct.

Linda: The move 2(8)2 would lose because Right could then play the pharisaic 7(9)8, creating a unicorn.

U.G.: A unicorn?

Linda: A sphere so simple that either player can force it to yield a single survivor.

U.G.: You call that a unicorn?

Linda: Or hermitic. Speaking of lingo, what did mean by "crashing a trap"?

U.G.: Crashing a trap is moving into a trap voluntarily. Sometimes moving into a type 2 trap by taking out the pivot can work out okay by favorably affecting two spheres at once. Your solution to my original problem is an example of successful trap crashing. But successful crashing is the exception rather than the rule.

Linda: I think you've taught me what I wanted to know. Now I'm going to go home and take your advice.

U.G.: Take my advice?

Linda: I'm going to analyze some of these positions for myself.

U.G.: Wait! If you want my phone number in case any...

Linda: Good luck with the twins!

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