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Guest Message by DevFuse

Analyze this: A simple card game

13 replies to this topic

#1 bonanova

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Posted 13 August 2012 - 06:58 PM

You must start with a deck that contains a triangular number of cards. From a standard deck, the largest such deck would contain 45 cards. You keep the cards face down throughout, so it does not matter which cards are selected.

The game begins by dealing the cards arbitrarily into a number of piles: anything from one pile of 45 [say] to 45 piles of a single card each. Play continues as a series of moves. A move consists of gathering one card from each pile to form a new pile. The piles are not in any particular order, but one card must be harvested from each pile during a move. Piles with only one card are destroyed, but each move creates a new pile.

The question concerns the end configuration, if one exists, after which any further moves do not change the number of piles nor change the number of cards in the collection of individual piles, without regard for which pile has what number of cards. i.e. four piles with 3 4 7 5 cards or 4 5 7 3 cards are considered the same configuration.

First, does the configuration eventually become stable? OK, there wouldn't be a second question if the answer were No; so as a mental exercise imagine, before trying, what the stable configuration would be. Then play a game to confirm it.

Second, using a deck of 45 cards, what is the maximum number of steps required to reach the stable configuration, and what initial configuration of piles produces it? It may be easier to work this out on paper using state trees than to deal the cards.

Extra credit: provide the second answer for all the triangular numbers possible with a standard deck.

Enjoy.
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Vidi vici veni.

#2 Yoruichi-san

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Posted 13 August 2012 - 07:10 PM

The first part is easy...
Spoiler for ...

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Women are definitely stronger. We are [Fe]males, after all...

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#3 curr3nt

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Posted 13 August 2012 - 07:37 PM

Spoiler for Max moves I've found...

Edited by curr3nt, 13 August 2012 - 07:42 PM.

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#4 CaptainEd

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Posted 13 August 2012 - 11:13 PM

Spoiler for My max steps so far

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#5 jim

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Posted 13 August 2012 - 11:22 PM

Question. If we start with 45 piles of one card each, do we immediately destroy all piles and finish with a configuation of no cards and no piles in no moves?
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#6 CaptainEd

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Posted 14 August 2012 - 12:27 AM

Jim, you harvest one from each of the piles, destroying them. However, your harvest becomes another pile, so you wind up with one pile with 45 cards. Then curr3nt has analyzed what happens there.

Spoiler for I can do 62

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#7 CaptainEd

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Posted 14 August 2012 - 06:46 AM

Spoiler for max steps and starting configs, by hand, up through T9=45

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#8 superprismatic

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Posted 14 August 2012 - 03:27 PM

Spoiler for I've exhausted through T11

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#9 CaptainEd

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Posted 14 August 2012 - 04:27 PM

Superprismatic, that's a pretty and nicely organized template for a starting configuration! This whole problem and its solution are provocative.
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#10 superprismatic

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Posted 14 August 2012 - 04:53 PM

Superprismatic, that's a pretty and nicely organized template for a starting configuration! This whole problem and its solution are provocative.

It's provocative alright! The trouble is, I have no idea how to proceed from here. I hope you have a glimmer of an idea, Captain.
By the way, I've seen this before somewhere. I think it was called Bulgarian Solitaire.
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