World
Game
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# Khorkov Announces Startling Findings

December 30, 2006

In a series of emails to fellow players, Roman Khorkov this week shared the results of work he has done recently on the misere game. These results are shocking. Roman has apparently overturned the Hudson conjecture, which has been the “prevailing wisdom” for the past two years and during two championships. If Roman is correct – and he is well-known to be an extremely thorough analyst – top players will have been choosing the wrong side to defend in selected openings all during this time period.

The Hudson conjecture was an alternative to the misere conjecture of Applegate, Jacobson and Sleator, who theorized that Left wins in misere sprouts when and only when n divided by 5 leaves a remainder of 0 or 1. The Hudson Conjecture claims that the true pattern only asserts itself when n is larger than 9. But for n > 9, Cal Hudson thought that Left wins in misere sprouts when and only when n divided by 6 leaves a remainder of 3, 4, or 5.

Cal Hudson, of course, never proved his conjecture, but he had a bit of a mechanism to explain it. He believed that with a sufficiently large value for n, any starting position must ultimately resolve into a UT plus an even or an odd number of survivors. Then, he claimed, the player winning in normal sprouts will also win in misere sprouts. A nice idea, but Roman evidently has analysis that overturns the supposed pattern and replaces it with a new pattern.

The new pattern, the Khorkov conjecture, says that Left wins when and only when n divided by 6 leaves a remainder of 4, 5, or 0.

In the series of emails, Roman also presented two general principles of misere play. We will give these verbatim and, rather than risk misinterpretation of the symbology, leave explication for future articles:

Gz= (x^y + (6n+2)spots)+ = (x^y + (6n+2)spots)-

Gz= (x^y + (6n+5)spots)+ = (x^y + (6n+5)spots)-

Roman has credited Josh Purinton with assisting him in his misere work.