A New Mathematical Principle - George Blake

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Joe Mckay
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A New Mathematical Principle - George Blake

Postby Joe Mckay » August 10th, 2009, 3:36 pm

In an earlier post, I mentioned that I have recently become very interested in the work of George Blake. This is due to a remarkable discovery he made in an old issue of 'The Pentagram' magazine. This was back in 1948 - so what I am about to mention is in no way new. But, since I have never seen it discussed before it has the feel of a new discovery. Anyway - the idea is called 'SINGLETON' and is detailed briefly on PAGE 23 of the DECEMBER 1948 issue of 'The Pentagram' magazine. A slight variation is then mentioned on PAGE 6 of the OCTOBER 1950 issue of the same magazine...

1) So - grab a deck of cards. Make sure it is a complete deck.

2) Give it as many shuffles as you want.

3) Now deal the deck into 4 heaps - BUT - misdeal on one heap so that one heap will have twelve, one fourteen and two thirteen cards...

4) Now reassemble the heaps and deal again without any misdeal.

6) Now if you look through each pile you are guaranteed to find a 'singleton' in at least one of the piles. A 'singleton' is the only card of its suit in that pile of cards (otherwise known as a 'mutton'in games of Bridge)... ie. There will be at leat one pile with either ONE club, heart, spade or diamond.

NOW - That is all there is to this. Obviously, it is a pretty unremarkable piece of magic (in fact it is more of a slight puzzle than an effect). But - I find it very intriguing. It is one of those things that continues to fool you even when you supposedly know how it is done. Why does this work? Does it work ALL the time or just MOST of the time? And how the hell did George discover such a screwy idea? It all reminds me of the sort of arcane trivia/principles that Stewart James would discover...

Has anybody else seen references to this idea in print? I am no mathematician (although somebody who is may be able to help explain this principle?) but it reminds me of 'The Pigeonhole Principle' which I have seen Martin Gardner discuss in one of his 'Mathematical Games' columns. I remember there were some cards tricks using this principle published in 'The Pallbearer's Review' a few decades ago. Annoyingly, I recently sold my set (gambling debts :cool: )... So - for now - I can't check out those effects which may or may not throw some light on this effect...

I believe this idea must be pretty obscure since it was only mentioned briefly in the 'Magic Go Round' section of 'The Pentagram' magazine (it is a kind of notes from the editor section). Perhaps it would have warranted more coverage if it had being published in its own right as a trick...

Would love to hear anything about the above - I am sure there is a small band of fellow magicians who find these mathematical curiosities/principles as fascinating as I do...

A tip of the hat to Mr Blake,

Joe

PS. In the variation (OCT 1950) the cards are dealt into 4 piles with a card being placed to one side. The cards are then gathered and re-dealt. When you go through the cards, a singleton is found. From the description, I am unsure as to where the card that is placed to one side should be returned when the cards are gathered up before the re-deal...

Curtis Kam
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Re: A New Mathematical Principle - George Blake

Postby Curtis Kam » August 10th, 2009, 3:50 pm

What are the odds of a singleton showing up in one of four piles without Blake's dealing procedure?

Joe Mckay
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Re: A New Mathematical Principle - George Blake

Postby Joe Mckay » August 10th, 2009, 5:37 pm

In the steps I listed above - I jump from step 4 to step 6. It is a simple mis-numbering by me. No need to worry that I have missed out a crucial step or anything... Sadly, after the ten minutes(?) are up it is too late to go back and edit the original post...

Joe

PS. I am not sure if the link to 'The Pigeonhole Principle' is accurate or not. But for those interested, some card effects using this principle also feature in 'Puzzler's Tribute: A Feast For The Mind'. This was one of the books published to honour the work of Martin Gardner... Actually - I have just found a link to the very effect featured in that book: http://www.spelman.edu/~colm/epstein.pdf

Larry Barnowsky
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Re: A New Mathematical Principle - George Blake

Postby Larry Barnowsky » August 10th, 2009, 6:25 pm

I would say the same odds.

This works quite frequently but not always.

A computer simulation program should be able to estimate the actual odds by brute force method.
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Denis Behr
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Re: A New Mathematical Principle - George Blake

Postby Denis Behr » August 10th, 2009, 7:00 pm

Joe Mckay wrote:2) Give it as many shuffles as you want.

3) Now deal the deck into 4 heaps - BUT - misdeal on one heap so that one heap will have twelve, one fourteen and two thirteen cards...

4) Now reassemble the heaps

Huh?! Are you sure you have that right? If you have a shuffled deck and do this dealing and then simply assemble the cards again, you still have a shuffled deck. Dealing the cards and collecting them doesn't do anything, so you can just skip it.

The assertion by Blake is, in other words: Just divide the deck in four equal piles in any way. One pile will have a singleton. Obviously, this is not true in all cases.

I guess I don't get it. Are you sure this was not in the April issue?

Denis

Matt Sedlak
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Re: A New Mathematical Principle - George Blake

Postby Matt Sedlak » August 10th, 2009, 7:31 pm

Counterexample: Start with the cards in CHSD order, in sets of 4 (i.e 4C,7H,AS,QD,6C,etc). Doesn't work. I don't have a copy of the original article so maybe you miscopied the procedure. I wasn't sure which way you meant (should I deal all the cards into a pile before moving on to the next or should I deal them into four piles simultaneously) so I tried both ways and it didn't work.

David Britland
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Re: A New Mathematical Principle - George Blake

Postby David Britland » August 10th, 2009, 8:40 pm

Hi Joe

I wrote about George Blake's Mysterious Mis-Deal in an article for Stan Allen's Magic Magazine called Tricks of Faith. It was about tricks that don't really work.

Although the idea was in the April 1954 issue of Max Andrews' Magic Magazine I think he believed it worked. Though Peter McDonald, writing in a subsequent issue said that ordinary deals produced the same effect.

As you say, Blake also had the idea in the Pentagram (after Peter Warlock relented and became convinced it might work) and, years later, in the Magigram, which is where I first saw it.

At the time I wrote that the chance of getting a singleton in a hand through a regular deal was about 73%, a figure that came from reading bridge books. But that figure is my own leap of faith!

Michael Close
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Re: A New Mathematical Principle - George Blake

Postby Michael Close » August 10th, 2009, 8:57 pm

Would that be that if you deal four hands the chance that one of them will contain a singleton will be 73%? The odds of any given hand being deal a singleton must be a much lower percentage.

Close (who is still fogged from FISM)

the Larry
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Re: A New Mathematical Principle - George Blake

Postby the Larry » August 11th, 2009, 2:21 am

I am not a mathematician but it is pretty obvious that this will not always work. Simply reverse the procedure. Start off with four piles without a singleton and then reverse the procedure. You will end up with a deck where this 'principle' will fail.

This is not more or less than dealing somebody four cards and stating "I bet you have two cards of the same suit". Chances are he has but it is not a guarantee.

David Britland
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Re: A New Mathematical Principle - George Blake

Postby David Britland » August 11th, 2009, 6:29 am

Hi Michael

Yes, it's the prediction that one of the hands will contain a singleton. But you don't know which one.

Carlo Morpurgo
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Re: A New Mathematical Principle - George Blake

Postby Carlo Morpurgo » August 11th, 2009, 11:31 am

Michael Close wrote:Would that be that if you deal four hands the chance that one of them will contain a singleton will be 73%? The odds of any given hand being deal a singleton must be a much lower percentage.

Close (who is still fogged from FISM)


for one hand it's not too bad, you have to count all possible groups of type 1-x-y-z, 1-1-x-y 1-1-1-z, where x,y,z are digits between 2 and 10, the sum of the digits in a given group is 13. For example to count groups 1-2-4-6 (one singleton in a suite, 2 card in a different suite etc.) you pick any of the 4!=24 choices of suites to assign to the 1,2,4 and 6. For each of those you have 13 choices for the "1", 13*12/2 for the "2", 13*12*11/3! for the "4", and 13*12*11*10*9*8/6! for the "6". The total is
the product of all that stuff which is 29858811840. This means that the probability of getting a singleton with 2-4-6 in other suites is that number divided by the total number of bridge hands which is 635013559600, and this gives ~0.047.

To get the probability of a singleton in a single hand do the above for all other groups and add up - I think you get 0.305, or 30.5%.

The calculation of all groups of 4 hands with at least one singleton seems a bit more involved.....

Carlo

Joe Mckay
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Re: A New Mathematical Principle - George Blake

Postby Joe Mckay » August 11th, 2009, 11:52 am

The article that David mentions is in the July 2002 issue of MAGIC. I am going to track it down and read it again. Obviously I have completely forgotten that David has written about this before. It isn't the first time that I have found something interesting, and then found that Mr Britland has written about it many years earlier! He is a fantastic writer and his website contains many hidden gems. You should all check out www.davidbritland.com

So - I guess this whole deal/mis-deal thing is a complete red herring. Apologies! I seemed to have being fooled by the similarites I thought I saw between this and some card effects using the Pigeonhole Principle. Martin Gardner writes about this in 'The Last Recreations'. I remember one of the effects, but can't remember if I read it in the Gardner book or in 'The Pallbearer's Review'. Perhaps variations ended up in both places. The basic idea was this:

1) Shuffle deck

2) Deal out five hands of cards (5 cards in each hand)

3) In each hand re-arrange the cards so that the highest valued card is placed first in the hand, the second highest valued card is placed second in the hand and so on.

4) Gather the cards and again deal them out into five hands.

5) Yet again you go through each hand and arrange the cards so that the highest valued card is placed first in the hand, the second highest valued card is placed second in the hand and so on...

6) Now - gather the cards and again deal them out into five hands.

7) And - for the last time - go through each hand and rearrange the cards in each hand just as you did on the previous two occasions.

8) Now - for the last time - deal out five hands of cards. You will find that each hand has now ordered themselves so that at each hand runs (from the top down) from the highest valued card to the lowest valued card.

9) No matter how many times you continue to deal out five hands you will find that each hand is still ordered from the highest to the lowest valued card... Pretty dull? Perhaps. But, pretty nonetheless...

If the above turns out to be wrong as well, I am going to give up completely. And take up rope magic.

Joe

Joe Mckay
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Re: A New Mathematical Principle - George Blake

Postby Joe Mckay » August 11th, 2009, 11:54 am

One other thing. Is it possible that this deal-misdeal procedure changes the odds just a little bit? Or is the procedure completely useless?

Joe

the Larry
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Re: A New Mathematical Principle - George Blake

Postby the Larry » August 11th, 2009, 5:17 pm

The procedure is completely useless.


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