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# Get the gold coin...

9 replies to this topic

### #1 Anza Power

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Posted 05 January 2013 - 12:57 PM

This was sent to me by a friend...

2 players play a game where an infinity 1-dimentional board is divided into cells numbered 0 1 2...

4 coins are placed on some cells in this board, the second coin from the left is gold:

(the position of the coins in this picture is just for show)

The players take turns, each player chooses one of the four coins and moves it 1 or more cells to the LEFT, you cannot surpass a coin or land on the same cell.

If you move a coin beyond the leftmost cell it is taken out of the game, whichever played takes out the gold coin wins...

So in the above example the player who's turn it is could either move the leftmost coin 1 cell or take it out by moving it 2 cells, or move the gold coin 1 or 2 cells, or move the third coin 1 cell, or move the fourth coin 1 2 or 3 cells...

Develop a strategy for winning the game given the initial positions of the coins...
• 0

### #2 bonanova

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Posted 07 January 2013 - 04:16 AM

Interesting puzzle!

Spoiler for Analogy

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Vidi vici veni.

### #3 bonanova

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Posted 07 January 2013 - 11:08 AM

Spoiler for Solution

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Vidi vici veni.

### #4 Anza Power

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Posted 07 January 2013 - 08:13 PM

^ bonanova excellent you got it, the game isn't about the general case, you know you only have 4 coins and the second one from the left is the golden one, only thing that can change is the position of these coins...

Your explanation is a bit lengthy though, all you need to do is:

Spoiler for

Edited by Anza Power, 07 January 2013 - 08:13 PM.

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### #5 bonanova

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Posted 07 January 2013 - 08:51 PM

First player always wins then?
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Vidi vici veni.

### #6 k-man

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Posted 07 January 2013 - 11:29 PM

First player always wins then?

Spoiler for Except...

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### #7 Prime

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Posted 08 January 2013 - 02:17 AM

^ bonanova excellent you got it, the game isn't about the general case, you know you only have 4 coins and the second one from the left is the golden one, only thing that can change is the position of these coins...

Your explanation is a bit lengthy though, all you need to do is:

Spoiler for

Spoiler for General rule?

• 0

Past prime, actually.

### #8 bonanova

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Posted 11 January 2013 - 09:00 AM

First player always wins then?

Spoiler for Except...

k-man, Great observation.   Anza Power, very interesting puzzle.

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Vidi vici veni.

### #9 Prime

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Posted 12 January 2013 - 11:17 AM

Interestingly enough, the XOR formula inhabits this game, like it does Nim.

Consider a bit more complex variation: two copper coins on the left, followed by gold coin, followed by two more copper coins.

Spoiler for XOR rules!

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Past prime, actually.

### #10 Prime

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Posted 14 January 2013 - 07:42 AM

A finishing touch...

Spoiler for The general formula

Armed with this knowledge, you could make a bet, whereby first your opponent sets up any position and then you choose who moves first.

• 1

Past prime, actually.

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