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Discovering integer sequences by dealing cards

card dealing

Let’s play a game:

  1. Imagine you have some playing cards. Of course if you actually have some cards you don’t need to imagine!
  2. Pick your favourite natural number $n$ and put a deck of $n$ cards in front of you. Then repeat the next step until the deck is empty.
  3. Take $2$ cards from the top of the deck and throw them away, or just take $1$ card from the top and throw it away. The choice is yours.

If you pick a small $n$, such as $n=3$, it’s pretty easy to see how this game is going to play out. Choosing to throw away $2$ cards the first time means you’re then forced to throw away $1$ card the next time, but only throwing away $1$ card the first time leaves you with a choice of what to throw away the next time. So for $n=3$ there are exactly $3$ different ways to play the game: throw $2$ then $1$, throw $1$ then $2$, or throw $1$ then $1$ then $1$.

Now, here comes the big question. How does the number of different ways to play this game depend on the size of the starting deck? Or in other words, what integer sequence $a_0$, $a_1$, $a_2$, $a_3$, $a_4$, … do we get if $a_n$ represents the number of different ways to play the game with a deck of $n$ cards? (We already know that $a_3=3$.)

Probability of dealing four perfect hands of cards in a world of random shufflers

A couple of months ago (really? Two years?! Man!) I posted about an extraordinary coincidence: in a game of whist at a village hall in Kineton, Warwickshire, each of four players had been dealt an entire suit each. My post ‘Four perfect hands: An event never seen before (right?)‘ discussed this story. What really interested me was that the quoted mathematical analysis — and figure of 2,235,197,406,895,366,368,301,559,999 to 1 — appears to be correct; what lets down the piece is poor modelling. The probability calculated relies on the assumption that the deck is completely randomly ordered. Apart from the fact that new decks of cards come sorted into suits, whist is a game of collecting like cards together, so a coincidental ordering must be made more likely. Still unlikely enough to be worthy of mention in a local paper, maybe, but not “this is the first time this hand has ever been dealt in the history of the game”-unlikely.

Anyway, last week I was asked where the quoted figure 2,235,197,406,895,366,368,301,559,999 to 1 actually comes from. Here’s my shot at it.

Card trick video from Christian Perfect

A while ago Christian Perfect suggested the monthly local Maths Jam organisers might write up what happens at Maths Jams to their blogs so others can get a feel for what goes on. I regard this as a good idea I haven’t got around to yet.

Luckily, Christian has just made a video showing a card trick we have played with at the Nottingham Maths Jam, so that makes this an easy post!

I was shown this trick by Matt Parker in a hotel bar in Coventry, who refused to say how it works. I went to the Nottingham Maths Jam in November 2011 having worked out how to do the trick but having spent no time at all considering how it might work, saving this for Maths Jam. I showed John Read, Kathryn Taylor and Sharon Evans and together we worked out the details given in Christian’s video.

I made a joke on Twitter based on Gauss’ reaction to Bolyai’s work on non-Euclidean geometry: “Enjoying video by @christianp. However, ‘to praise it would amount to praising myself’ ;)”. Gauss is reported to have written to Bolyai’s father:

To praise [Bolyai’s work] would amount to praising myself. For the entire content of the work … coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years.

while privately writing to a friend to say:

I regard this young geometer Bolyai as a genius of the first order.

Of course, by invoking the former I meant to imply the latter. Perhaps a more suitable quote might be that of Kelvin, having first read George Green’s Essay on electricity and magnetism:

I have just met with Green’s memoir, which renders a separate treatise on electricity less necessary… I have, most unwittingly, trodden almost exactly in his steps as far as regards electricity.

I’d say playing around with tricks and working out how they work is a very Maths Jam activity so anyone considering attending one should regard this as very much the sort of thing that happens at a Maths Jam. Find your local one, or set one up!

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