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John and Jane are playing a game. They both go to the bank and each get 1000 euros in two-euro pieces. So each of them has 500 coins at stake. They then find a large rectangle table and decide that whoever wins the following game gets to keep all of the coins currently on the table. The rules are as follows:

  • It is a turn-based game, so John and Jane take turns alternately (1 coin per turn).
  • The coins are placed on the table and they may not be put on top of each other.
  • Coins may not be moved once put on the table.
  • Assume the table isn't large enough to hold all the coins.
  • If there isn't any room to put coins on the table, the person who placed the last possible coin wins.

What is the winning game strategy?

Community
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Kevin
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  • How many coins can they place per turn? – Stormenet Oct 06 '14 at 11:47
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    without knowing the size of the table it is impossible to know – Elgert Oct 06 '14 at 12:16
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    There also seems to be a vocabulary issue here: "Get 1000 euros in 2 euros, so each has 500 euros at state". I don't know the terminology, but this problem seems simplifiable to "500 coins of equal size and value". – Mooing Duck Oct 06 '14 at 19:07
  • What if the table is sufficiently large to place all coins? Does player 2, as the last player to place a coin, win the challenge? What if the table is non-symmetric? What if gravity fails and the coins don't stay on the table? We need answers! – FreeAsInBeer Oct 06 '14 at 19:22
  • On a big enough table the second player wins, as the first player runs out of coins first. – Helena Aug 29 '23 at 08:21

1 Answers1

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The winning strategy for the first player is to put their coin in the dead center of the table. Then whatever move their opponent makes, they exactly mirror it, around the center.

e.g. If the second player puts their first coin 1 inch to the left of the center coin, the first player mirrors this by putting their coin 1 inch to the right of the center coin.

It's relatively easy to see why this works. If a spot is free for player 2, then its mirroring spot must also be free for player 1 because after each turn player 1 takes, there will be no unmirrored spots left. This means that wherever player 2 goes, there's guaranteed to be a spot left for player 1. So the only person who can possibly reach a state where there's no spot to go is player 2, who inevitably loses (unless they didn't bring enough coins to cover their table!)

Ben Aaronson
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  • gratz! that was fast. +1 – Rafe Oct 06 '14 at 12:16
  • Exactly! Great job! – Kevin Oct 06 '14 at 12:39
  • What about the second player (lets just assume that the first player is not aware of the winning strategy for player 1) – Kristoffer Sall-Storgaard Oct 06 '14 at 13:45
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    @KristofferSHansen: I am not aware of a proof of who wins if the first player does not play at the center. I suspect it depends on the dimensions of the table (measured in coin diameters). – Ross Millikan Oct 06 '14 at 14:36
  • This solution also relies on the ability of player one to be able to determine the exact centre of the table with a tolerance max of the diameter of one of the coins, which may be quite difficult. (If they are out by more than one coin, then there won't be space for the mirror in the outer layers) –  Oct 07 '14 at 13:50
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    @Stacey If I could win significant amount of money all I need is a measuring tape and I would measure it all out. – Kevin Oct 07 '14 at 15:07
  • How can we prove this to an 8th grader ? That is, what is a simple proof that the first player will always be able to place his coin without overlap ? – Hemant Agarwal Jun 19 '21 at 11:45
  • @HemantAgarwal, I think the current proof as is is already quite simple to follow. Which part do you think an 8th grader will have trouble understanding? – justhalf Jun 23 '21 at 03:54