The magician puts out hardly a deck of $52$ cards and announces that $51$ of them will be thrown out of the table, and there will remain three of clubs. The viewer at each step says which count from the edge the card should be thrown out, and the magician chooses to count from the left or right edge, and ejects the corresponding card. At what initial positions of the three of clubs can the success of the focus be guaranteed?
Problem
Source: International Mathematical Tournament of towns
Tags: combinatorics
cooljoseph
03.12.2019 01:39
The answer is that the card must be on one of the edges. These positions clearly work. To show that a position in the middle does not work, replace the viewer with a robot that always says, "Throw out the second card." Then the magician cannot ever remove an edge card until there are only $2$ cards left, meaning the last card must be one of the edges.
skrublord420
03.12.2019 03:14
Great solution and great problem. I probably should've not accidentally spoiled the solution for myself.
Jackshi2006
03.12.2019 03:21
I don't understand this sentence: The viewer at each step says which count from the edge the card should be thrown out, and the magician chooses to count from the left or right edge, and ejects the corresponding card. Does this mean the viewer can choose how many cards in to throw out, and the magician can choose which side to count in from?
skrublord420
03.12.2019 03:37
The viewer yells a number out, say $7,$ and then the magician either removes the $7$th card from the left or the $7$th card from the right.