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.
(In reply to
Some thoughts by Jer)
Not sure this works, but my thought is something like this:
Message 1: Dealer chooses the cards with values 3 and 1. Call them A and B. Player simply knows that card A beats card B, but nothing else.
Message 2: Dealer chooses the cards with values 3 and 2. Call the latter card C. The player knows that A beats C, and now knows that the value of card A must be at least 3, since it can beat two other cards.
Message 3: Dealer shows that Card D (value = 4) beats Card A. Therefore player knows card D must be at least 4.
Message 4: Dealer shows that Card E (value = 5) beats Card D. Therefore Card E >= 5.
...
Message 99: Dealer chooses the cards with values 100 and 99. At this point, the player will know that the former must be greater than or equal to 100, and since 100 is the max, they know it is the card with 100. He can now work his way backward through all the cards all the way down to Card A.
Message 100: Dealer shows that C beats B, which tells the player that C is 2 and B is 1.
EDIT: Nah, I see my mistake, obviously at message 3 the player knows the card is at least 4, OR is the card with value 1.
Edited on March 4, 2021, 11:44 am
Edited on March 4, 2021, 11:45 am
|
|
Posted by tomarken
on 2021-03-04 11:39:49 |
re: Some thoughts
