Jump to content


Welcome to BrainDen.com - Brain Teasers Forum

Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account.
As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends.

Of course, you can also enjoy our collection of amazing optical illusions and cool math games.

If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top.
If you have a website, we would appreciate a little link to BrainDen.

Thanks and enjoy the Den :-)
Guest Message by DevFuse
 

Photo
* * * * - 1 votes

Analyze this: A simple card game


  • Please log in to reply
13 replies to this topic

#1 bonanova

bonanova

    bonanova

  • Moderator
  • PipPipPipPip
  • 5776 posts
  • Gender:Male
  • Location:New York

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. ;)
  • 0
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#2 Yoruichi-san

Yoruichi-san

    "That Woman"

  • Members
  • PipPipPipPip
  • 3377 posts
  • Gender:Not Telling
  • Location:Outside the box

Posted 13 August 2012 - 07:10 PM

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

  • 0
Women are definitely stronger. We are [Fe]males, after all...

Some of what makes me me is real, some of what makes me me is imaginary...I guess I'm just complex. ;P

<3 BBC's Sherlock, the series and the man. "Smart is the new sexy."

Chromatic Witch links now on my 'About Me' page! Episode 3 is finally here!

When life hands me lemons, I make invisible ink.

#3 curr3nt

curr3nt

    The

  • Members
  • PipPipPipPip
  • 2839 posts
  • Gender:Male
  • Location:in a field of spatulas...

Posted 13 August 2012 - 07:37 PM

Spoiler for Max moves I've found...

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

  • 0

#4 CaptainEd

CaptainEd

    Senior Member

  • Members
  • PipPipPipPip
  • 1094 posts

Posted 13 August 2012 - 11:13 PM

Spoiler for My max steps so far

  • 0

#5 jim

jim

    Junior Member

  • Members
  • PipPip
  • 33 posts

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?
  • 0

#6 CaptainEd

CaptainEd

    Senior Member

  • Members
  • PipPipPipPip
  • 1094 posts

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

  • 0

#7 CaptainEd

CaptainEd

    Senior Member

  • Members
  • PipPipPipPip
  • 1094 posts

Posted 14 August 2012 - 06:46 AM

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

  • 0

#8 superprismatic

superprismatic

    Not just Prismatic

  • Moderator
  • PipPipPipPip
  • 1281 posts
  • Gender:Male

Posted 14 August 2012 - 03:27 PM

Spoiler for I've exhausted through T11

  • 0

#9 CaptainEd

CaptainEd

    Senior Member

  • Members
  • PipPipPipPip
  • 1094 posts

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.
  • 0

#10 superprismatic

superprismatic

    Not just Prismatic

  • Moderator
  • PipPipPipPip
  • 1281 posts
  • Gender:Male

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.
  • 0




0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users