DV. Dealing with Chaos

May 5, 2008 · The Mathcast, answers, logic · Permalink

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We explore Barry Cipra’s Tag Deal a bit more…

 
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4 Comments »

  1. nklein said,

    May 5, 2008 at 4:29 pm

    I just wanted to add another reason this makes a very bad bar bet: any misdeal gets you out of the good circuit. Chaim said two players with 52 cards takes 7 deals. But, if two cards stick together unnoticed on deal five, you’ve got to hope for some very special misdeals in the future to get you back on track.

    There are 53 ways to divide the cards between two people. Only 7 of those ways are on the good loop. You don’t want to end up in the other 46.

    (It is only 7 and not 14, right? because it matters who is dealing…)

  2. trivial34 said,

    May 5, 2008 at 9:07 pm

    I have been working on this problem daily since I heard the first show on the wonderful game of Tag Deal. I have a few interesting observations to report along with my progress. I noticed on the show and in the comments that people are trying to find a pattern for a given number of players with a varying number of cards. I do not believe this is the right way to look at this. If we look at a given number of cards and vary the number of players, the cycle length is much more orderly. In fact, for k cards and p>=k-1 players the cycle lengths form an arithmetic sequence with a common difference that is always a power of 2. This common difference will be 2^Ceiling(log_2 k) where k is the number of cards. This has given me hope that a formula for at least all cycle lengths of k cards and p>=k-1 players exists. Since we can calculate these differences we almost have a formula. All we need to find is a formula which gives us the cycle length for k cards and k-1 players (which I call the stabilized values) and then add the correct number of differences. The sequence of stabilized values is even more surprising. Making a table of differences between these values (and then a table of the differences between those!) shows a really nice pattern forming. I don’t have the proper notation or typeset to display the differences here but I’m sure if you make a table you’ll see the pattern. The patterns are recursive which is giving me trouble but I think I am getting very close to a formula. Let me know what anyone can make of these patterns.

  3. nklein said,

    May 7, 2008 at 6:45 pm

    Ugh. I cannot figure out how to put an image inline here in the comments.
    So, I’m just going to point you to my LiveJournal.

  4. djogon said,

    May 9, 2008 at 10:38 am

    I created a little program that calculates the tags, saves the states of each player and the graphically represents them. It may help, but it is highly amusing to see how the number of required tag varies with different number of players/cards.

    I planned to create more graphical representations that may help “discover” a patter or at least look pretty :)

    Anyway…
    You can pick it up from
    http://www.softwareriver.com/download/orderlychaos.zip

    You only need to have .NET 2.x on your computer and it should run. Select the number of cards, players and select start.
    You will then see a graphical representation of each tag as a line for each player.
    The player with no cards will have an empty line - the player with all cards will have the full line for that tag (row).
    Starting player is colored blue.

    If the “graph” cannot fit in your window a red line will show up indicating that there is more so try to resize your window.

    Have fun!

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