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GOOD JUDGMENT

by John McCarthy, Computer Science Department, Stanford University

(A mathematics fiction story dedicated to L. E. J. Brouwer)

	Once a spaceship containing a gang of would-be conquistadores
and a virtuous anthropologist found a planet where there were certainly
natives and a good chance of valuable ore.

	The conquistadores started checking out the ore, and the
anthropologist (whom regulations had required they bring along) started
checking out the natives.

	Our anthropologist (call him Oliver) found a native culture
of a common peaceful sort with only one significant peculiarity.  The
natives believed in accurate judgment as a cardinal virtue, emphasized
it in all their dealings and taught it to their children.  For example,
a young girl would be asked to cut some flowers and put them on the
dinner table in vase.  She would do it but then her mother would emphasize
getting the vase in just the right place so that everyone could see it
equally.  The vase might be moved back and forth twenty times before
mother and daughter agreed that it was placed just right.  Oliver couldn't
see how it made much difference, but he noted that the older children
were always better able to please their elders about placing things
just right.

	Meanwhile the conquistadores found their ore and decided that the
natives should mine it for them.  Our hero unsuccessfully cited regulations
against exploitation but could not dissuade them.  His conscience overcame
his loyalty (rather weak) to his fellow Earthmen and he decided to warn
the natives and persuade them to flee to the hills.

	They listened to him carefully, but asked only one question.  "How
many conquistadores were there?"  When told there were twenty they relaxed,
and our hero could not convince them that the weapons of the conquistadores
far more than made up for their small number.

	Oliver could not understand the native preparations for the
coming battle.  They went to the woods and selected small trees which
were chopped down, incidentally making good use of the axes they had
been given to insure their initial co-operation with the explorations
of the conquistadores.

	The final results of the native labors were strange: 20 stepladders
20 posts pointed on the bottoms, and 20 sledgehammers.  Oliver attempted
to convince them that these could not possibly prevail against the
conquistadores rayguns that could paralyze a man at 200 yards.
The complacent natives seemed entirely satisfied with their preparations
and went back to their usual activities, although Oliver noticed that their
children no longer ventured far in the direction of the pass over which
was parked the space ship.

	Finally the dread day arrived.  The bad guys came over the pass,
visible several miles away, and the natives went into action.  They formed
twenty teams of three.  One member of each team carried the step ladder,
a second the post and the third the sledge hammer.  They advanced a few
hundred yards from the village after which was the usual shouting back and
forth about what was the correct placement.  After dithering for fifteen
minutes, they came to an agreement.  One man held the step ladder while
another climbed it with the sledge hammer.  The third man held the post
and moved it about till they agreed that it was in exactly the right
place.  Then the third man held it while the second man drove it into
the ground with the sledge hammer.  After this they all retired to the
village with the ladders and hammers.

	They made our hero keep his head down, but he could still see
the curious event that occurred.  Each conquistadore found himself in
front of a stake unable to decide whether to pass it on the right or
on the left.  An hour later they were still undecided.  Once one said,
"Let's all go to the right, but this was just as they were all so close
to going to the left (including himself) that it was just sufficient
to bring themselves back to the point of indecision.

	Night fell and they were still undecided.  The next morning
came without result.  Once in while a native, after consultation
with the others, would show himself to the bad guys, apparently
in just such a way as would insure their continued inability to decide.
Finally the natives buried them where they fell.

Problem: Formulate and prove a theorem showing that there is a placement
of the stakes that would do what the natives needed.

Remark: Oliver's report of this event caused great excitement on Earth,
but subsequent investigations provided no clue to the natives' ability.
"They just seem to be lucky" was the only conclusion the scientists could
reach.