Scott, the more cards there are the less a certain possible match is affected by if there are other matches in the deck (it _is_ affected and that's where your thinking went wrong). The fewer cards there are the more it's affected as you'll easily see with thought experiments with decks of only 2 cards or 3 cards, just as Jack showed you if you bothered reading his post close to the beginning of the thread. That's why your wrong formula converges to the correct solution as the number of cards goes infinite.

/Tomas


The series oscillates around 1/e so the next term that is added or subtracted will land the sum on the other side of 1/e, thus you can sharpen that result by saying that it's actually within 1/(n+1)! of 1/e.

/Tomas

Following landmarks example and TomasB's suggestion to consider small decks:

1 card decks, prob of no match = (1-1)/1=0, therefore prob of match =1 (sure thing), that's correct.

2 card decks, on 1st card probability of no match = (2-1)/2=1/2. Bingo. If it didn't match on the first card it won't match on the second either. So the result for the second card is not independent of the first. The probabilities don't multiply in this case. The correct answer is 1/2 that there will be a match (and if there is one there will be two).

3 card decks, on 1st card no match = 2/3, on 2nd ditto, and on 3rd ditto. 2/3x2/3x2/3 = 0.296 (Scott's approach). Correct answer by enumeration =0.333. That is, there are two examples with no match out of six possibilities. The probability of at least one match is 4/6.

Scott's solution is approached as the number of cards gets large, but breaks down when the number of cards is small.

Now will you guys please stop trying to drive me nuts? (It's a short drive).

Steve

I'd like to get some collective help with deciding the composition of the decks in the effect I described earlier:

"Two marked decks are shuffled. A third deck is still in its case. Cards are dealt face up and face down until a face matches the writing on a back. This might happen on count 23. When the cards are turned over they show a totally different face (maybe 3 of Clubs) and a totally different writing (maybe "Seven of Spades"). The third deck is uncased and the 23rd down card is found to be the 3 of Clubs with "Seven of Spades" written on its back."

The properties I desire (but am sure can not be fulfilled are): Spectator decides which deck he gets and which deck you get. He decides if he deals face up or face down. He decides the same for you. He decides if the third cased deck is dealt face up or face down in the end. Most importantly, in the case of no match during the first deal you should not need to reverse shuffle your whole deck for the re-deal.

All these can probably not be met, but which ones are most important?

The free choice of deck is handled if at a certain position deck 1 has the card (back/face) A/B, deck 2 has B/C and deck 3 has C/A.

The most symmetric compilation of the decks I've found to be if you add the two jokers, shuffle the deck then number the cards as 0 through 53. Each deck then has identical cards where the cards have the identities (back/face) A/(A+18)mod54. Cut the decks so that deck 1 has card 0 written on top, deck 2 has card 18 written on top and deck 3 has card 36 written on top. This compilation has some nice properties:

Easy to re-stack a shuffled deck since you just copy any of the unshuffled decks and cut it. Any choice of deck for you and any choice of deck for the spectator. (Unfortunately I can't see that he can choose who deals face up and who deals face down.) The third cased deck should be dealt in the same orientation you deal yours. If a re-deal is required maybe it's possible to just deal with the deck in the other orientation? Or just deal from top and deal the third deck in the other orientation.

Any other ideas for stacks with more desirable properties or is this enough?

/Tomas

The principle being discussed in this thread was found by George Kaplan and the trick using it was described by J.G.Thomson in his My Best (1945). I forgot the exact name of the trick. I tried to find the book in the mess of my library, but failed. Maybe it was Coincidence or Provisions? Kaplan's trick is very interesting as the Invisible Deck is used to show the third coincidence.
Hideo Kato

P.S.
As I am not good at mathematics. I once simulated by computer and got 64% as the result.

To: Hideo Kato

You may not feel you are good at mathematics, but your memory is in great shape.

The George Kaplan effect is called "Coincidence? No. Prevision? Yes". It is on pages 94-95 of "My Best".

Steve

An earlier reference would be The Sphinx Vol. 39, No. 3, p. 67 in May 1940 where Tom Bowyer published this as "The Frequent Miracle".

Seems as if Ted Lesley reinvented Kaplan's idea of combining this with an Invisible Deck when he published "The More Frequent Miracle" in Paramiracles.

/Tomas

Thanks sashain for your clarifying. And thanks TomasB for additional information.

I think Tom Bowyer's name was credited in "My Best". sashain, could you clarify it for us as mess of my library can't be tidied up in a few days?

Hideo Kato

P.S.
I just posted a trick influenced by the members' posts in this thread in Secret Session.

To: Hideo Kato

Yes, Tom Bowyer is mentioned in “My Best”, (pg. 95.) However, the specific reference to the Sphinx (which is given by TomasB, above) is not mentioned.

Steve

Thanks again sashain for the clarification.

As TomasB gave above important information, I checked "The Frequent Miracle" in The Sphinx Vol. 39, No. 3, May 1940. By reading the writing by Tom Bowyer, we can confirm that he found the high probability of matching in dealing two decks of cards simultaneously.

So the first post of mine in this thread is including a false. George Kaplan created the trick based on the principle Tom Bowyer discovered.

I found another trick by Peter Kane based on the same principle titled My Best Coincidence in his Another Card Session published in 1971.

Just during writing this post, I created another trick with this principle. Maybe it will be posted in 30 minutes.

Hideo Kato

Mr Kato,

I can't wait to get access to the Secret Session forum.

/Tomas

Hello Tomas, while you posted above, I have posted 2 more tricks (totally 3) using the principle we are discussing. Hurry up!

Hideo Kato

My addition, as far as I know, to this principle is to keep one or more decks stacked to get a known result connected to the random position of the match. Does anyone know if this has been done in combination with the Frequent Miracle method before? If not...I think I'll call the combination of principles "Frequent Miracle Stack".

I also think I added the alternate dealing procedure which in effect pushes the probability of a match up to the same level as if you had dealt through the decks twice.

As a side-note: If you combine the alternate dealing with letting one spectator cut off the top portion of a deck and shuffle it while the second spectator shuffles the bottom portion, before they assemble the deck, you in effect deal through a deck FOUR times during one deal which makes for a 98% success rate.

/Tomas

Tomas, I'm not understanding this. How does letting the spectator shuffle affect the probability? Can you be more specific about exactly what is happening here? I can't picture what you're saying, how does this combine with the alternate dealing? Sorry for being so dense.

Thanks,
Jack Shalom

It was just me describing it unclearly. Keep two decks identically stacked and shuffle the top portions separately from the bottom portions. If you deal the top portions simultaneously you in fact have two 26-card decks where the probability of a match is 63%. If you deal the bottom portions simultaneously you have a match 63% of the time there also. Putting the top portions on the bottom portions before the deal will therefore lead to a success rate of 86%.

Combining that with the alternate dealing procedure will push it up to 98%.

Any other ideas to increase the probability of a match?

/Tomas

Hey, I was doing the effect originally with 63% (I like to take risks) bringing it up to 98% is pretty safe to me... I say, no need to go further.
back to the presentational point of view. for those who use a nail writer I would also suggest to do the effect with 2 specs, and to nail write the card or position (or both although that is too much to NW) when you discover a match. they will be so busy doing the dealing and "wanting" it to succeed, it is practically no problem to nail write.

Wow! That should be declared illegal it's so powerful. Thanks Tomas.

Jack Shalom

Please refer my posts in Secret Session about this principle. I used an alternate dealing, matching with stacked deck and shifting 1 card to adjust.

Congratulations Tomas for arriving at 50 posts.

Hideo Kato

Tomas,

Your idea with dividing the deck gave me an idea on how to make this effect even stronger (IMHO.)

Have 2 decks divided in 2 portions. The decks need not be in the same order but just make sure the top half of each deck contains the same cards (which leads to the fact that the bottom half of both decks will contain the same cards as well.)
If working with a few spectators, divide the decks in half and give specs to shuffle. Assemble the decks back so one keeps its original top and bottom half, but the other is reversed in such a way that the top half will go on bottom and vice versa.
Demonstrate that it the chances are "astronomical" to get a match by dealing the two decks.
Actually it is impossible when the decks are constructed as described.
Have the halves shuffled again, this time assemble the decks the right way.
Show how easy it is for you to find a match.

Of course, the presentational aspect needs strengthening but you guys get the idea.
It is basically an extension of Tomas' idea but the other way around.

I hope you guys like it.

Nir

Just saw this thread… interesting stuff.

Since the two halves are being kept separate during the shuffling procedure, do make sure that each half has equal representation of all of the card values. Otherwise certain card combinations will have a higher chance of hitting and still others will have a lower chance of hitting (I think...).

This might be leveraged to advantage. The deck could be arranged so certain combinations will *never* hit and increase the chances for others. If all the 6's are in the bottom half and the 9's are all in the top half, they will never match. You could demonstrate how unlikely a hit is this way and also totally exclude the 9/6 pairing from their choice as its the one you did in demonstrating the procedure.

It would add considerable time to an effect that could drag if it isn't presented strongly.

My $.02...

- Greg Owen

Greg, if you choose the route with shuffling each (approximate) half separately, what you write will be automatic. If they cut off 23 cards, of course the top 23 cards in both decks will consist of the same 23 cards.

As for Nir's suggestion I think that that presentation can be saved for when you genuinely get no match although you wanted to get one. It's therefore important to before the deal speak of how highly improbable it is to get the exact same card in the same position in both decks. If randomness works against you during the first deal it just confirms what you just said.

/Tomas