In a card game, each card is associated with a numerical value from 1 to 100, with each card beating less, with one exception: 1 beats 100. The player knows that 100 cards with different values lie in front of him. The dealer who knows the order of these cards can tell the player which card beats the other for any pair of cards he draws. Prove that the dealer can make one hundred such messages, so that after that the player can accurately determine the value of each card.
Ok, 2nd attempt, similar to my first:
Msg 1: Dealer shows that A beats B.
Msg 2: Dealer shows that B beats C.
Msg 3: Dealer shows that C beats A.
We now know that one of these cards is 1, one of them is 100, and the third is somewhere in between.
Msg 4: Dealer shows that C beats D. We now know that C > D > 1. Either A = 1 and B = 100, or B = 1 and C = 100.
Msg 5: Dealer shows that E beats C. We now know that E > C > D > 1. A = 1 and B = 100.
Msg 6: Dealer shows that F beats E. Therefore F >= 5.
Msg 7: Dealer shows that G beats F. Therefore G >= 6.
Msg 8: Dealer shows that H beats G. Therefore H >= 7.
...
Msg 98: Dealer shows that X beats W. Therefore X >= 97.
Msg 99: Dealer shows that Y beats X. Therefore Y >= 98.
Msg 100: Dealer shows that Z beats Y. Therefore Z >= 99.
But we already know from Msg 5 that B = 100, therefore Z = 99, and then Y = 98, etc. all the way down to C = 3, D = 2.
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Posted by tomarken
on 2021-03-10 15:09:45 |
Solution?
