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| XGID=--AC--A------------n------:0:0:1:61:2:4:1:5:10 | |||||
Where is the missing White checker? | |||||
eXtreme Gammon Version: 2.10
As he contemplated his play, Benjamin's cell phone rang, and he had to rush off to take care of some emergency.
His opponent left the position as it was, and wrote down the score: He had 4 points and Benjamin had 2 (in a match to 5), Crawford. Then, to pass the time, he went over to watch the football game about to start. He knelt with everyone else as the national anthem played, then sat at the bar and ordered a drink.
Benjamin returned to the cocktail lounge a couple of hours later, found his opponent, and they went back to the table to continue their match. The position was still as they left it, except there were only 14 White checkers (see above diagram). The 15th lay beside the board, which Benjamin noted was none other than the big White button he had used to replace the White checker he had lost the previous night.
The bartender, aware of the commotion, welcomed Benjamin back and explained. "Shortly after you left, I noticed a pair of gentlemen sitting at the very chairs you left vacant. One of them was waving something around, speaking animatedly and gesturing at the board. I went over and informed them that, while I understood that the button was quite a conversation piece, there was a backgammon match in progress, and asked them when they were finished to please put back the button where they had found it.
The man holding the button looked sheepish. 'That's just it. We don't remember where I picked it up from, only that it was somewhere in this half of the board, in other words in either Blue's home board or White's home board.' "They were genuinely apologetic, though they had no further clues to offer me.
"However, being something of a backgammon enthusiast myself, the position intrigued me for two reasons:
"First, Blue's checker position is identical to the one that arose when you asked me last night if I had seen your missing White checker.
"Secondly, I wondered about Blue's best play with 61 and if it might depend on the location of your button" (he pointed to the White checker misplaced on the side of the board). "I did 4-ply rollouts on XG for each of the inner board placements, until I had reached 100% confidence and always at least 5k trials.
"What I discovered is that, according to the rollouts, there are three different best plays for Blue, depending on the placement of the button. Moreover, there is only one location at which each of these candidates is solely best. For the rest of the locations, two plays are exactly tied -- long rollouts converge them to exactly 50%."
What are the three locations?
Nack