World
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The Last Match of 2007 - Complete!

24- (Purvis-Peltier*) 1(25)1[2-6]

As recommended by Roman early last year. I hope the theory has not changed since then! - Danny Purvis

..... 2(26)3

5L18 seems a good move which is not in my table, probably a 2^0. I seek to destroy the firmer part composed of the 5 spots. - Jeff Peltier

1(27)25[2]

This move should assure that the sphere S(27) is "shallow" (no subsequent UTs) and of even parity (an even number of survivors). I still have not mastered the standard misere lingo and thus do not understand Jeff's use of the word "firmer" above. - Danny

First "fickle" is a game which is not a switch (I use your terms) and for which the parity of survivors is different in normal or misere and so has genus 0^1 or 1^0. A firm component is an attractor as Firm+Firm=Firm and Fickle+Firm=Firm. Examples are the UT (0^0) and the 5-Spots game (1^1), they have usual genus x^x. Usually sum of a game G + firm component is firm, a counter-example in 7 Spots is known in case of sum with the UT (discovered by Roman). No counter-example is yet known with 5-Spots component. - Jeff

Is "attractor" a technical term? Has Roman's counterexample been published previously? - Danny

No attractor is mine. Will you add fickle and firm to WGOSA glossary as I think they are basic bricks of genus theory and Sprouts is very peculiar for its large set of fickle positions namely 3n? - Jeff

Excellent suggestion! I agree. Will do! - Danny

I was referring to this article for the counterexample of sum of a UT + a game with a not firm result. - Jeff

Back to the game. Why do I want the sphere S(27) to be a shallow even? I am simply struggling to follow Roman's original plan, which I only vaguely understand. But in general I need overall a shallow even or a deep odd. So now I will need the sphere S(7) to be a shallow even or a deep odd. - Danny

Would you reconsider Danny (I hope you mean 1(27)1[2,3,26])? As clearly after 26(28)27 there will not be an even number of survivors but just 3 of them? - Jeff

..... 26(28)27

Quite right. I blundered and will now probably lose this match. But of course I will not reconsider, and I have played for you the move you mention. Let me know if you want to play something different. If not I will now play 7(29)8, hoping for complications. - Danny

7(29)8 7(30)7[8,9] This sort of move has no real theory yet, just copied from Roman's games, I hope I understand it one day. -Jeff

Since 15- has been solved (though I have not seen the analysis) I will not steer that way. Instead, I'll try for a different double switches. (But then your new double switches weapon will likely give me a gloomy future here also!) - Danny

10(31)30 11(32)31

My table says that I can transform the PP15 of genus 3^4+ into a 6L8 of genus 2^2 to match the switch of value 3^3 (PP2) + 1^0 for the 3 spots [4-6] -Jeff

I suspect my only hope is to steer for complications. However, I doubt that can be accomplished without Right's cooperation. - Danny

12(33)13 8(34)8

I think the component you played (Double-Anago in A15) is 1^1 so I match it with a 1^0 from the 3 Spots [4-6] and I let you enjoy complexity ! - Jeff

14(35)15

Once again I misvisualized the position, and I have been pondering the wrong position for the past three days. I will simply hope my move, necessarily made with almost no thought, is complicated enough for me still to have chances. - Danny

..... 10(36)10[11,16]

Romanian way to simplify a position by enclosing a spot and an anago. - Jeff

17(37)18 12(38)33[14,19-21,36]

Still in the fog - Jeff

It's an interesting struggle. Of the two roughly symmetrical spheres, I need both to be shallow or else I need to break the parity symmetry. - Danny

12(39)38[19]

Yes, they are very shallow or in genus language deep, at least 0^3 against 1^5. Now I'll try to get *2 to be added to the *1 to match the 0^3 component : - Jeff

..... 17(40)18[22]

You seem to be giving me the bad news that my word "deep" is already taken. Speaking of genus language, I have not yet had time to study Josh's exposition of that subject, but I hope to soon. But am I right to infer that for any classification i ^ j, for integers i and j, that j could be replaced meaningfully by an infinite, repeating sequence of integers? -- Danny

Well j is the misere nim value and if you want complete genus, the following digits will be misere nim value of the game you get when you add to game g a nim heap of size 2 each time: So genus of g is G+(g)^G-(g)G-(g+*2)G-(g+*2+*2).... which becomes periodic and usually pretty soon. That is why Genus theory in incomplete because it says nearly nothing of say the value of g+*4. -- Jeff

I have a lot to learn. -- Danny

Me too, in a recent discussion with Yper I just realized the difference between two simple games *2+*2 and *4+*4 both of same genus 0^020... -- Jeff

Back to the game... I have found a move that still allows me a modicum of hope or delusion. I will happily play it without further meditation, in a sort of sprouts Zen spirit, which I dedicate to the Dalai Lama and oppressed citizens of Tibet. -- Danny

14(41)14[20,21]

Coming back to a nice x^3 component and 0^0 in S(15) . - Jeff

..... 32(42)36

After much analysis, I seem to have determined that the sphere S(17) is a minor deity, with pretensions at least to the class of gods of which a person typically is able to find an exemplar in every golden oyster. Sad for me, this impressive entity refuses to yield the two or four switchmen that I need. Therefore, I resign. -- Danny

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Congratulations! -- Danny

Thanks Danny, and sorry to have badly played move 32 and to have made you think I played 6L8, so despite its complexity it is not a game to keep for next generations of Sprouts players. -- Jeff

21- (Purvis*-Peltier) 1(22)2 The standard move, trying to make a presumably losing position complicated. - Danny 1(23)1[2,3] The only answer known to me - Jeff.

4(24)23 Now I take solace in the complexity of double switches. - Danny 7(25)7[5,6,24] An invention of mine I test for the first time, I don't know of any good reply but I might be surprised, I hope to have hit a 3^3 to match your PP2 switch of genus 3^3. - Jeff

What a nightmare! Every time I think the future of competitive sprouts is secure, a move like this pops up! Congratulations on finding such a promising weapon against double switches! - Danny

7(26)25[8-13] 4(27)24

I thought for some time you played the awful 5P9 of unknown genus 1^4+ (like 5P3) which might have been a winning move to which I would have tried same answer... (Please Roman or Josh say what you think after the tournament). But no you played the cheerful 6P8 of genus 1^1, so now I have two options : either convert the switches PP3+3^3 to the right value or convert the 1^1 of 6P8 to the right value. For sake of simplicity I chose first solution here 2^1+3^3=1^x, but I let you compute the value --Jeff

I did not even look at the 5P9. I wish I had! - Danny

2(28)22 8(29)26

Here I hope that 2^2+2^1+PP6+8Spots is 2^2+2^1+4^1+0^0 and yields 4^0 ! This is a nice try as 0L2 + 0L1 is not a nice animal at all with options of normal Grundy values only G2 and G3...and so I can not find a G1 to match 6P8 there. --Jeff

I don't understand why you evaluate the sphere S(4) as a 2^1. I think it is a 1^2. - Danny

Two simple options making it at least 2^1 are UT of genus 0^0 and UT type 2 of genus 1^0. - Jeff

Thanks! I can see now that it is a 2^1. But even with this help, I don't know how to manipulate these misere Grundy numbers. After reading the recent correspondence and Josh's new, as-yet-unpublished article on the subject, I might be able to handle fickle and firm biospheres, but don't think 2^1 or 1^2 would be either fickle or firm. - Danny

Right! - Jeff

Is this where the mysterious Noah's Ark theorem comes into play? - Danny

No it would be useful only if you take two restive animals (for example 1^2) that would be equivalent to a firm 0^0 or two restless animals (for example 2^1) the pair being equivalent to a fickle 0: 0^1. - Jeff

I made an effort to understand the sphere S(8), but found it too difficult for me. I looked at your last-published chart (which I had vowed not to do) and noted that you give pp6 a 4^1. I currently cannot translate 4^1 into my semantic universe with much precision, but the evaluation definitely influences me to reject one of my two candidate moves. So now I play the other. I am currently optimistic about the resulting position, which I evaluate as (approximately) double switches with even switchmen. - Danny

I think you are right, what I really need now is a *4 (of genus 4^4), the current position is a bit too firm (firm options) so I'll use one of my ways to go toward more fickle types. - Jeff

8(30)8[9-13] 9(31)9[10-13]

Exactly the move I most feared. I have spent much time over this position in the past three days, and I have become increasingly gloomy regarding it. Nevertheless, I cannot see quite to the bottom. - Danny

9(32@30)31 30(33)32

The move I expected (reverting the previous 5^3 sphere to wild 0^1), now I must be careful because 4-spots components are too wild to be caught by simplified genus or nim values - Jeff

10(34)10[11]

I have been very reluctant to make that move, but now I think it might be my best chance. I am hoping that you will be forced to create another switch and that I can then toggle that switch favorably. - Danny

..... 14(35)14[15-17]

Yes, I still don't know how this will end - Jeff

Two hours after making my previous comment, I discovered a flaw in my analysis. When you create another switch, you will in effect be creating two switches at once, because the sphere S(34) will suddenly become switchlike. There will then be, approximately, an even number of switches, disallowng an immediate toggle. When I had that thought I wanted to write you again, proffering an immediate resignation. But later I recollected that sphere S(34) could then continue to evolve, becoming a true switch and with the parity I then would need. So then I concluded, again, that I am actually winning. I have not had the time or the heart since to make a new diagram and give the position more study. (I always seem to lose my old diagrams between moves. I desperately need to become much more organized.) But I will play with happy simplicity, dreaming of the Dalai Lama, of the true meaning of the Olympic torch, and of victory... -- Danny

14(36)35[15]

Here I was amazed you did not chose the famous Roman's game 0P3. I think there is no simple answer here but I don't want to disclose too much the reason for this PP2 of value *3 - Jeff

..... 16(37)36

After rioting several hours in the back alleys of our other game, I am in no mood for more thinking. So I will rely on the plan I outlined in my previous note without further evaluation. -- Danny

12(38)13

At move 37 we have :

S4 + 0L1 + 0L2+ 1L2 + PP3
supposed genus
1⁰²⁰² + *2 + 2¹⁴¹ + 0³ + *3

after Danny's move 38 we get

S4 + 0L1 + 0L2 + 1LA2 + PP3
1⁰²⁰² + *2 + 2¹⁴¹ + *2 + *3

where amazingly
1⁰²⁰² + *2 + 2¹⁴¹ + *2 = 3^0..
hence my move 39 reverting the *3 PP2 to *0, but this may need some checking... due to the misere mysteries even for a Dalai Lama : --Jeff

............ 16(39)16

What if Danny had played S4 to A4 of genus 3³, then :
3³ + *2 + 2¹⁴¹ + 0³ + *3, I think this may be 0⁰·· --Jeff

Good move. I can see now that I'm lost. (18(40)18[19] 20(41)21 20(42)21 4(43)27 P 18(40)18[19] 20(41)21 18(42)41 4(43)27[5] P 2(40)28 12(41)13 4(42)27 18(43)18[19] 20(44)21 5(45)6 P 4(40)27 18(41)18[19] P 4(40)27[5] 17(41)39 18(42)18[19] 2(43)3 P) Thanks for the games, or I should say, thanks for the lessons! -- Danny

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