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.

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## bonanova

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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