Magic trick based on deep mathematics I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to choose two integers between 1 and 50 and add them. Then add the largest two of the three integers at hand. Then add the largest two again. Repeat this around ten times. Alice tells the magician her final number $n$. The magician then tells Alice the next number. This is done by computing $(1.61803398\cdots) n$ and rounding to the nearest integer. The explanation is beyond the comprehension of a random mathematical layperson, but for a mathematician it is not very deep. Can anyone do better?
 A: I gave a talk about card shuffling to a general audience recently and wanted to memorise a "random-looking" deck so as to motivate a correct definition of what it means for a deck to be random. Most magicians actually use memory tricks to learn off the deck but I thought it would be much cleverer to order the cards in the obvious way, and then find a recursive sequence of length 52 containing all of 1 to 52. In the end, caught for time I settled on using the Collatz recursive relation with seed 18 --- this allowed me to name off 21 distinct cards effortlessly and when I held up the deck prior to the demonstration, the audience voted that the deck was random. Can anyone think of a suitable recursive sequence with the desired property? We can either take a random-looking order and a "regular" recursive sequence but I think it would be much better to find an easy to compute recursive sequence that "looks random" when using a more canonical order simply because if we can remember a "random looking order" we're pretty much going to have to remember the whole deck --- the problem I'm exactly trying to avoid.
PS: I did one of the simpler Diaconis tricks. A deck is riffle shuffled three times, the top card shown to the audience, inserted into the deck, and after laying the cards out on the table the top card can be easily recovered by looking at the descents. The key is that the order of the deck is known beforehand --- a simple demonstration that three shuffles does not suffice to mix up a deck of cards (with respect to variation distance).
A: Here is a simple trick based on group theory.  Ask a person to choose four numbers from 1 to 9 and write them in a row on a piece of paper.  Pause for a moment and then write a number on a piece of paper without letting the other person see what it is.  Turn the paper over and place it on the table.
Now ask the person to choose two of the numbers from the list and put a line though them.  Ask the person to compute a*b + a + b and put it in the list to replace the two chosen numbers. 
Continue to do this until there is only one remaining number.  Turn over the paper and show that the numbers match.
The simplest way of explaining this is to show that a * b + a + b is isomorphic to multiplication using the transform T(x) = x + 1.  (a*b + a + b) + 1 = (a + 1)(b + 1).  If we denote the operation a * b + a + b as a & b, this means that a & b is commuative and associative, just as multiplication is.  For any list of numbers ai,  the final number can be computed as the (a1 + 1)(a2 + 1)...(an + 1) - 1.
A: Ask someone to lay out the 52 cards in a deck, face up, in 4 rows of 13 cards each, in any order the person wants. Then you can always pick 13 cards, one from each column, in such a way as to get exactly one card of each denomination (that is, one ace, one deuce, ..., one king). 
As a trick, it's not up there with sawing a woman in half, but its explanation does require Hall's Marriage Theorem. 
A: Two persons, A and B, perform this trick. The public (or one from the public) chooses two natural numbers and give A the sum and B the product. A and B will ask each other, alternatively, the only single question "Do you know the numbers?" answering only yes or no until both find the numbers. There is a strategy such that for any input and only doing this, A and B will manage to find the original numbers. 
I have never seen magicians actually performing this, but is perfectly doable. 
This was a problem in the shortlist of the proposed problem for some international mathematical olympiad. Unfortunately I don't remember which. If someone remembers or finds it. Tell us please. i would also like to know. 
A: Here's another Fibonacci trick, from Benjamin & Quinn's "Proofs that really count".

The magician hands a volunteer a sheet of paper with a table whose rows are numbered from one to ten, plus a final row for the total. She asks him to fill in the first two rows with his favorite two positive integers. She then asks him to fill in row three with the sum of the first two rows, row four with the sum of row two and row three, etcetera... She then hands him a calculator and asks him to add up all ten numbers together. Before he's able to finish that, the magician has a quick look at the sheet of paper and announces the total. The magician then asks the volunteer to divide row 10 by row 9, and cut up the answer to the second decimal digit. The volunteer performs the division and says: 1.61. And the magician: "Now turn over the paper and look what I've written". The paper says: "I predict the number 1.61".

The first part of the trick uses the following well-known Fibonacci identity:
$$\sum_{i=1}^nF_i=F_{n+2}-1$$
Indeed, call $x$ the number in row 1 and $y$ the number in row 2. Then for $n \geq 3$, the number in row $n$ is $F_{n-2} x+F_{n-1} y$, where $F_n$ is the $n$-th Fibonacci number. So the number in row 7 is $F_5 x + F_6 y=5x+8y$ and the total is $$x+y+\sum_{i=3}^{10} (F_{i-2} x+F_{i-1} y)= F_{10} x + F_{11} y=55x+88y$$ by the Fibonacci identity mentioned at the beginning. Therefore all the magician has to do to find the total is multiply row 7 by the number 11.
The second part of the trick uses an inequality for the freshman sum ;-) of two fractions. That is, given positive fractions $\frac{a}{b}$ and $\frac{c}{d}$ such that $\frac{a}{b}<\frac{c}{d}$ we have:
$$\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$$
Just note that the number in row 9 is $13x+21y$ while the number in row 10 is $21x+34y$. Hence:
$$
1.615 \dots =\frac{21x}{13x} < \frac{21x+34y}{13x+21y} < \frac{34y}{21y}=1.619 \dots
$$
A: This was fascinating for me. Somehow the man takes a bagel and with one cut arrives with two pieces that are interlocked. Whether this qualifies as "magic" I dunno (it's hard to say once the trick's been explained), but it sure seems like it to me.
It doesn't hurt that I love bagels, and have the opportunity to perform this with friends/family/non-math people and can teach a little about problems/topology/counter-intuitive facts about the universe.
A: Here's an example of a magic trick that works with high probability, based on a careful analysis of the riffle shuffle, in which an audience member performs a number of riffle shuffles and then moves a single card, and the magician guesses which card has been moved.
A: Not so much a magic trick as a math trick, in that I can prove it works in theory but I have never tried it in practice.
Take a very long one-dimensional frictionless billiard table, with a wall at one end.  Away from the wall, place a billiard ball with mass $10^{2n}$ for $n$ positive.  Between that ball and the wall, place another billiard ball with mass $1$.  Then start the heavy ball rolling slowly towards the light one.  Of course, they bounce, setting the light one traveling quickly towards the wall, which it bounces off, and then it hits the heavy ball, etc., until all the momentum from the heavy ball has been transferred and it starts rolling away.
Assume that all collisions are perfectly elastic.  Then at the end of the day, there will be finitely many collisions.  Indeed, the number of collisions will calculate the digits of $\pi$, in the sense that there will be $\lfloor \pi \times 10^n \rfloor$ collisions.
I prefer this method of calculating $\pi$ much better than the probabilistic one.
A: So two points of note.
I did not read all the posts above in detail but did do a search for the Faro Shuffle and got no results... So:
This is a shuffle where all the cards interweave absolutely perfectly (so a perfect riffle shuffle). There's quite a lot of maths behind this. For instance, 8 shuffles takes you back to the order you started shuffling the cards in. Martin Gardner talked about this a bit in at least one of his SA columns. The problem with the faro shuffle is it takes a long long time to learn... personally well over a year, and that was with the benefit of having been a practicing amateur magician for along time. Still if interested the book to look for is The Collected Works of Alex Elmsley, this really lays the foundations for mathematical faro work...
Another trick I came across whilst working towards an Ergodic Theory exam uses the Birkhoff Ergodic Theorem at its core. You can read about it in these notes: http://www.maths.manchester.ac.uk/~cwalkden/ergodic-theory/lecture22.pdf
Owen.
A: Peter Suber writes:
By the way, the single best knot trick I've ever found is at pp. 98-99 of Louis Kauffman's On Knots, a mathematical treatise listed below with the books on knot theory. I'm sure you've seen the trick in which someone ties an overhand knot by crossing their arms before picking up the cord, and then uncrossing them. Kauffman shows you how to do the same trick without crossing your arms first. The version of this trick in Ashley #2576 and Budworth 1977 [p. 151] is not nearly as good.
Work out how it is possible for yourselves! A link to the book is here.
[Edit: This magic trick does not rely on mathematics -- instead it violates an important mathematical fact, that the trefoil is not unknotted! The Chinese rings have a similar feel, but the mathematics violated (linking number) is less deep.]
A: 
Place $K$ faced-down cards on a table, blindfold yourself and ask him/her for a number $1 < n < K$. Allow him/her to flip $n$ random cards up. Cover the cards with an opaque box that has two holes for you to put your hands in and claim that you can split the cards into 2 stacks, each with same number of faced-up cards.  

Based on a well known logic puzzle: http://usna.edu/Users/physics/mungan/_files/documents/Scholarship/CoinPuzzle.pdf 
Modified the process to make it harder for audience to figure out what you did and used cards so that they will not think that you did it by differentiating the surface of the coins.
A: A recent episode of Penn and Teller's Fool Us had a trick by Hans Petter Secker which exploits the parity of a permutation in a lovely way.  It may be useful the next time one has to teach the sign of a permutation!
Brief description of the trick in the video:  The magician has sent a box over.  Penn picks up a Rock from the box; Teller then picks Scissors from the box; and Alysson gets the remaining crumpled sheet of paper.  The magician (on video) says that he will predict what they'll do (in the context of the Rock-Paper-Scissors game).  He writes down a prediction, and invites Alysson to pick any two of them and swap their objects. The magician reveals his correct prediction.  Next he writes down two predictions, and asks Teller to perform a swap, and then Penn to perform a swap (I might be forgetting who selects the swaps, but two swaps are made).  Again the magician reveals his correct predictions.  Finally he gets them to make three swaps.  This time he has not written down any predictions, but when the crumpled paper is smoothed out, there are three correct predictions written on it. Enjoy!
https://www.youtube.com/watch?app=desktop&v=hf7Sy7FPal0
A: Five unrelated items:
Mobius strip
One of the best mathematical tricks is what happens when you cut a Mobius strip in the middle. (Look here) (And what happens when you cut it again, and when you cut it not in the middle.) This is truly mind boggling and magicians use it in their acts. And it reflects deep mathematics.
Diaconis mind reading trick
I also heard from Mark Gorseky this description of a mathematical based card game
"Mark described a card trick of Diaconis where he takes a deck of cards, gives it to a person at the end of the room, lets this person “cut” the deck and replace the two parts, then asks many other people do the same and then asks people to take one card each from the deck. Next Diaconis is trying to read the mind of the five people with the last cards by asking them to concentrate on the cards they have. To help him a little against noise coming from other minds he asks those with black cards to step forward. Then he guesses the cards each of the five people have.
Mark said that Diaconis likes to perform this magic with a crowd of magician since it violates the basic rule: “never let the cards out of your control”. This trick is performed (with a reduced deck of 32 cards) based on a simple linear feedback shift register. Since all the operations of cuting and pasting amount to cyclic permutations, the 5 red/black bits are enough to tell the cylic shift and no genuine mind reading is required."
I think there is a paper by Goresky and Klapper about a version of this magic and relations to shift registers.
The Link Illusion
I heard a wonderful magic from Nahva De Shalit. You tie a string between the two hands of
two people and link the two strings. The task is to get unlinked.
This ties with what I heard from Eric Demaine about the main principle behined many puzzles (Some of which he manufectured with his father whan he was six!)
Symmetry Illusion
Sometimes things are not as symmetric as they may look.
commutators-based magic
(I heard this from Eric Demaine and from Shahar Mozes.) If we hang a picture (or boxing gloves) with one nail, once the nail falls so does the picture. If we use two nails then ordinarily if one nails falls the picture can still hangs there. Mathematics can come for the rescue for the following important task: use five nails so that if any one nail falls so does the picture.

A: I saw this trick demonstrated at a math camp once. When it works, it is extremely impressive to non-mathematicians and mathematicians alike.
Have a volunteer shuffle a deck of cards, select a card, show it to the audience, and shuffle it back into the deck. Take the deck from him, and fling all of the cards into the air. Grab one as it falls, and ask the volunteer if it is his card.
1 in 52 times (this is the deep mathematics part), the card you grab will be the card the volunteer selected. Even most statisticians should be amazed at this feat. Just make sure you never perform this trick twice to the same audience.
A: I forgot the historical name for this and I'm pretty sure this is classical and well-known. 
Consider a circular disk and remove an interior circular region, not necessarily concentric. In this annulus we play the following game. Start at any point $p_{1}$ of the outer boundary and draw a line through this point which is tangent to the inner circle. This line intersects the outer circle at another point $p_2$. Now repeat the same procedure with $p_2$ and get $p_3$. Iterating this procedure ad infinitum we either conclude that these sequence of points are periodic or not. What's true is that the periodicity or lack of it is independent of the starting point $p_1$. 
I believe there is a proof involving Lefschetz fixed point theorem involving the torus but any details on this and the history of this is more than welcome.
A: Magician: "Here is a deck of 27 cards. Select one, memorize it, put it back and shuffle at libitum. Now name a number between 1 and 27 inclusive (=: N)." Then the magician deals the cards face up into three heaps. You have to tell him in which heap the selected card lies, and he quickly ramasses the three heaps. This is done three times, then he hands you the deck, and you have to count N cards from its back. The N'th card is flipped over, and it turns out to be the card you have originally selected. 
A: This is a trick that I designed years ago and I have used it in many different occasions for amusement only or educational purpose or both. It is indeed the finial difference method to find a polynomial. Ask the person to write down a polynomial without you knowing the polynomial  and even the degree of the polynomial. To keep your life easy, it would be better to keep the degree less than or equal 3. (It wouldn't be hard to let a layman know what a polynomial is just by giving two or three examples). Then you ask for some information that is essentially the value of the polynomial for 0, 1, 2, 3. As soon as you take one of the value you should calculate the difference. And in a few seconds after taking the last information, you announce not only the degree of the polynomial but also the exact polynomial. 
Note 1: Finding the degree is a very important part of this trick since it convinces more knowlegable persons that you are not just solving a simultaneous equation quickly. 
Note 2: I used this trick in my Calculus classes to give this seemingly paradoxical idea that "if you don't know what the function is, try to figure out how it changes." 
Note 3: Of course, one can use it in many different classes for different purposes.
Note 4: I've just search the internet to see if Martin Gardner ever introduced this trick. Damn it! The answer was yes, here: "The calculus of finite differences". However, I still love to keep the credit of telling the degree for my self :)  
A: My favourite example, the rope and two carabiner trick
http://blogs.scientificamerican.com/guest-blog/amazing-rope-trick/
I also offer up my variation of the Dirac belt trick
https://m.youtube.com/watch?v=UtdljdoFAwg
A: Here is a trick much in the spirit of the original number-adding example; moreover I'm sure Richard will appreciate the type of "deep mathematics" involved.
On a rectangular board of a given size $m\times n$, Alice places (in absence of the magician) the numbers $1$ to $mn$ (written on cards) in such a way that rows and columns are increasing but otherwise at random (in math term she chooses a random rectangular standard Young tableau). She also chooses one of the numbers say $k$ and records its place on the board. Now the she removes the number $1$ at the top left and fills the empty square by a "jeu de taquin" sequence of moves (each time the empty square is filled from the right or from below, choosing the smaller candidate to keep rows and columns increasing, and until no candidates are left). This is repeated for the number $2$ (now at the top left) and so forth until $k-1$ is gone and $k$ is at the top left. Now enters the magician, looks at the board briefly, and then points out the original position of $k$ that Alice had recorded. For maximum surprise $k$ should be chosen away from the extremities of the range, and certainly not $1$ or $mn$ whose original positions are obvious.
All the magician needs to do is mentally determine the path the next slide (removing $k$) would take, and apply a central symmetry with respect to the center of the rectangle to the final square of that path.
In fact, the magician could in principle locate the original squares of all remaining numbers (but probably not mentally), simply by continuing to apply jeu de taquin slides. The fact that the tableau shown to the magician determines the original positions of all remaining numbers can be understood from the relatively well known properties of invertibility and confluence of jeu de taquin: one could slide back all remaining numbers to the bottom right corner, choosing the slides in an arbitrary order. However that would be virtually impossible to do mentally. The fact that the described simple method works is based on the less known fact that the Schútzenberger dual of any rectangular tableau can be obtained by negating the entries and applying central symmetry (see the final page of my contribution to the Foata Festschrift).
A: Persi Diaconis and Ron Graham just published Magical Mathematics. The book contains a plethora of magic tricks rooted in deep mathematics.
A: The following trick uses some relatively deep mathematics, namely cluster algebras.  It will probably impress (some) mathematicians, but not very many laypeople.
Draw a triangular grid and place 1s in some two rows, like the following except you may vary the distance between the 1s:
1   1   1   1   1   1   1
  .   .   .   .   .   .
.   .   .   .   .   .   .
  .   .   .   .   .   .
.   .   .   .   .   .   .
  .   .   .   .   .   .
1   1   1   1   1   1   1

Now choose some path from the top row of 1s to the bottom row and fill it in with 1s also, like so:
1   1   1   1   1   1   1
  1   .   .   .   .   .
.   1   .   .   .   .   .
  1   .   .   .   .   .
.   1   .   .   .   .   .
  .   1   .   .   .   .
1   1   1   1   1   1   1

Finally, fill in all of the entries of the grid with a number such that for every 2 by 2 "subsquare"
  b
a   d
  c

the condition $ad-bc=1$ is satisfied, or equivalently, that $d=\frac{bc+1}{a}$.  You can easily do this locally, filling in one forced entry after another.  For example, one might get the following:
1   1   1   1   1   1   1
  1   2   3   2   2   1
.   1   5   5   3   1   .
  1   2   8   7   1   .
.   1   3   11  2   1   .
  .   1   4   3   1   .
1   1   1   1   1   1   1

The "trick" is that every entry is an integer, and that the pattern of 1s quickly repeats, except upside-down.  If you were to continue to the right (and left), then you would have an infinite repeating pattern.
This should seem at least a bit surprising at first because you sometimes divide some fairly large numbers, e.g. $\frac{5\cdot 11+1}{8} = 7$ or $\frac{7\cdot 3+1}{11} = 2$ in the above picture.  Of course, the larger the grid you made initially, the larger the numbers will be, and the more surprising the exact division will be.
Incidentally, if anyone can provide a reference as to why this all works, I'd love to see it.  I managed to prove that all of the entries are integers, and that they're bounded, and so there will eventually be repetition.  However, the repetition distance is actually a simple function of the distance between the two rows of 1, which I can't prove.
A: The Kruskal count.
A: You may ask the person to encode something by RSA, then you decode it (you have the private key)
OR
To divide two 40-digit integers and give you the decimal result to 100 digits, you then use continued fractions to find the original fraction (reduced)
OR
To compute pq and pr where p,q,r are prime, you then find p,q,r by the Euclidean algorithm (no very deep, but it's the best i've got)
A: Here's a couple of well-known simple topology tricks:
Tie ends of a long enough piece of rope to your wrists, while wearing a loosely fitting jacket or sweatshirt. With your arms tied like that, take the jacket off your back and put it back on inside out. It's easier to figure out how to do it than to explain it in words, so I'll skip the explanation. The more risque version is to tie the ankles and do the trick with pants.
The other one I haven't tried, but maybe it can be done at a party if you have a stick and some plasticine around.
http://www.youtube.com/watch?v=S5fPwE7GQOA
A: A variant on  Anton Geraschenko answer above- say you are in a fourth grade school that for some reason let these poor kids use calculators. you ask them to pick for themselves a 3 digit number say abc. Tell them to write it twice in their calculator ,i.e., abcabc and then divide by 77. Then by 13. What did you get? do it again with 143 and then by 7? What did you get. again with...
It teaches them about prime decomposition, about the decimal structure, about consecutive division etc.
I learnt it from Avraham Arcavi.
A: Destination Unknown is a magic trick that makes use of Combinatorics. It really fools people. 
See http://themagicwarehouse.com/cgi-bin/findit.pl?x_item=SP2453&keyword=DESTINATION 
A: Although one of the answers mentions Kruskal count, I would like to add more about it. Kruskal count not just works in the case of a paragraph as well. If you start from one of the first ten words of a paragraph, go to the word which is away from the previous word by number of words exactly equal to the letters of the previous word, you'll land up on the same word in the end! For more discussion, you can refer this link
A: A late addition: The Fold and One-Cut Theorem.  Any straight-line drawing 
on a sheet of paper may be folded flat so that, with one straight scissors cut right through the paper,
exactly the drawing falls out, and nothing else. Houdini's 1922 book Paper Magic includes instructions
on how to cut out a 5-point star with one cut.  Martin Gardner posed the general question in his Scientific American column in 1960.  
For the proof, see Chapter 17 of Geometric Folding Algorithms: Linkages, Origami, Polyhedra.
We include instructions for cutting out a turtle, which, in my experience, draws a gasp from the audience. :-)
A: The coffee mug trick
Give a coffee mug (full if you're brave) to someone and ask them to rotate 360 degrees without spilling the (real or imaginary) coffee, so that their hand ends up in the same position.
This is impossible, so you get to smirk while they contort themselves and become more and more baffled (this works better with more than one person since it turns into a kind of "competition")
Finally, take the cup and show that while it's impossible to turn it once (as has been "proven"), it's possible to turn it twice (!) and end up in the same position.
Has to do with the fundamental group of SO(3) being $\mathbb{Z}/2\mathbb{Z}$, and when we require the cup to stay upright we end with a non-trivial loop.
A: Apart from tricks based on numbers, there are topological objects whose properties can seem quite magical, like the Möbius strip or the unknot.   
E.g. take a standard page of paper, show that it has two sides (number them with a pen, show that any straight pen path meets a boundary). Next, cut out a long strip from it (not needed of course, but adds to the drama), and ask the audience "and how many sides does this have?". They reply "two". Then you put the the two small ends of the strip together to form a ring and you ask "and now, how many sides?", they still reply "two!". At this point do a little diversion, like putting a pair of scissors on the table saying out loud "I'll use this in a minute".  Now do a half-twist with the strip before putting the small ends together and ask again "for the last time people, how many sides?". They answer "twoo!!", and you say "the magic has worked people, there's only one side!"  (you show that now the pen paths along the long direction never meet a boundary and come back). Most laymen are quite bemused. Now do two half-twists and ask again, some won't dare an answer...
A: If you are not mathematically inclined, this game can drive you crazy.
http://www.transience.com.au/pearl.html
A: 
Here is a general trick that you can use to make yourself look like you have an amazing memory.

Start with a finite abelian group $(G,+)$ in which you are comfortable doing arithmetic. Be sure to know the sum $$g^* = \sum_{g \in G} g.$$
Take a set $S$ of $|G|$ physical objects with an easily computable set isomorphism 
$$ \varphi : S \longrightarrow G.$$ 
Allow your audience to remove one random element from $a \in S$ and then shuffle $S$ without telling you what $a$ is. [Shuffling means we need $G$ to be abelian.]
Now inform your audience that you are going to look briefly at each remaining element of $S$ and remember exactly which elements you saw, and determine by process of elimination which element of $S$ was removed.
Now glace through all the remaining elements of $S$ one by one and keep a "running total" to compute
$$ \varphi(a) = g^* - \sum_{s \in S-\{a\}} \varphi(s).$$
Finally apply $\varphi^{-1}$ and obtain $a.$
Note that $\varphi$ is not "canonical" in the sense there are definitely choices to be made. On the other hand in should be "natural" in the sense that you should be very comfortable saying $s = \varphi(s).$

The prototypical example is to take $G$ to be $\Bbb Z / 13 \Bbb Z \times V_4,$ $S$ to be a standard deck of 52 cards, and $\varphi(s)$ to be $( \text{rank}(s) , \text{suit}(s) )$.

A: "The best card trick", an article by Michael Kleber.  Here is the opening paragraph:
"You, my friend, are about to witness
the best card trick there is.
Here, take this ordinary deck of cards,
and draw a hand of five cards from
it. Choose them deliberately or randomly,
whichever you prefer--but do
not show them to me! Show them instead
to my lovely assistant, who will
now give me four of them: the 7 of spades,
then the Q of hearts, the 8 of clubs, the 3 of diamonds. 
There is one card left in your hand, known
only to you and my assistant. And the
hidden card, my friend, is the K of spades."
A: Audience asked to choose an integer from 0 to 1000. Ask to give remainder when divided by 7, 11, and 13 respectively. 
Magician gives original integer by Chinese Remainder Theorem.
Works because 7×11×13=1001.
A: This trick exploits the thinness of coins.
http://www.howtodotricks.com/easy-coin-magic-trick.html
A: A puzzle based on Hamming codes discussed here: is the following:
A room contains a normal 8×8 chess board together with 64 identical coins, each with one “heads” side and one “tails” side. Two prisoners are at the mercy of a typically eccentric jailer who has decided to play a game with them for their freedom. The rules are the game are as follows.
The jailer will take one of the prisoners (let us call him the “first” prisoner) with him into the aforementioned room, leaving the second prisoner outside. Inside the room the jailer will place exactly one coin on each square of the chess board, choosing to show heads or tails as he sees fit (e.g. randomly). Having done this he will then choose one square of the chess board and declare to the first prisoner that this is the “magic” square. The first prisoner must then turn over exactly one of the coins and exit the room. After the first prisoner has left the room, the second prisoner is admitted. The jailer will ask him to identify the magic square. If he is able to do this, both prisoners will be granted their freedom
A: You can use hamming codes to guess a number with lying allowed. For example, here is a way to guess a number 0-15 with 7 yes-or-no questions, and the person being questioned is allowed to lie once. (The full cards are here).
A: I would like to thank all the contributors on this page.  I have been putting together a new Math-a-Magic show for the 9-12 grade level and have found some fantastic material here.  If I get a decent video of the show I'll be sure to post a link here so you can play the "What concept is behind this trick?" game.
I have modified some of your ideas severely.  For example. Craig Feinstein's suggestion was 
a commercial effect that asks the volunteer to pick one of a hundred different cities typed out on ten cards.  The volunteer finds the city's name on two different cards which the magician looks at casually.  You can then instantly tell him the name of the city he has mentally picked.
In my version, I instruct him not to ever let me see the city's name on the cards and yet I can still easily predict his choice!
Here is my favorite trick based on the deep principal of Set theory.  Ok, maybe it is not too deep but the results are astounding!  
Taking a deck of cards, you mention you have a prediction about these cards.  That means  it is very important to give the cards a really random shuffle.
You then give your volunteer half the deck and you both shuffle your half decks thoroughly.  Tell your volunteer to take a small amount(5-15) of cards from his half of the deck, turn them upside down and give them to you.  you do the same to him (doesn't matter how many cards you turn upside down as long as there is some left in your hand)
You then both shuffle the cards you have received into the deck in your hands in there upside down state.  so at this point both people will have some cards right side up some cards upside down.  You will follow the same procedure two more times.  It doesn't matter how many or what cards he or you are turning upside down and giving away. After all this is done put both half decks of cards together again (IMPORTANT: turn your entire half deck over when you place it on top of his)
Now you spread the cards out across a table top.  They should be a seemingly random mix of upside down and rigtside up cards. You then unfold your prediction slip which says something like:  11 cards will be black, 15 cards will be red, 6 will be clubs and 5 will be hearts and the hearts will also form a royal Flush!
They will be astounded by your amazingly detailed prediction.  What happens is that all the face up cards are the ones that were originally in your half deck.  This trick is self working.  All you do is to pick out which cards you want in your half of the deck and place them at the top of the deck to start.  Then just give him the random bottom half of the deck and you keep the pre-set ones.
Any questions?  Just email me at kevin@hallsofmagic.com
A: Here is a card trick from Edwin Connell's Elements of Abstract and Linear Algebra, page 18 (it can be found online).  I always do this trick to my undergraduate number theory class in the first minutes of the first day.  A few weeks later, after they've learned some modular arithmetic, we come back to the trick to see why it works.  I quote from Connell:
"Ask friends to pick out seven cards from a deck and then to select one
to look at without showing it to you. Take the six cards face down in your left hand
and the selected card in your right hand, and announce you will place the selected
card in with the other six, but they are not to know where. Put your hands behind
your back and place the selected card on top, and bring the seven cards in front in
your left hand. Ask your friends to give you a number between one and seven (not
allowing one). Suppose they say three. You move the top card to the bottom, then
the second card to the bottom, and then you turn over the third card, leaving it face
up on top. Then repeat the process, moving the top two cards to the bottom and
turning the third card face up on top. Continue until there is only one card face
down, and this will be the selected card."
When I do this trick, I always use big magician's cards (much easier for an audience to see), but a regular deck works too.  To get to the trick faster, I skip the first part and just pick 7 cards myself, showing them all the cards so they see nothing is funny (like two ace of spades or something).  I then spread the cards in one hand face-down and let a student pick one and show it to everyone else but me before I take it back face down.  When the student is showing the cards to the class I move the rest of the cards behind me so that before I get the card back I already have the rest behind my back.  
You need to make sure students at the side of the room won't be able to see what you're doing behind your back (namely, putting the mystery card on the top of the deck), so stand close to the board.  Practice this with yourself many times first to be sure you can do it without screwing up. The hard part is remembering to keep the last card you reached in the count on the top of the deck; that same card will be used when you start the count in the next round.  If you stick it on the bottom before counting off cards again then you'll mess everything up.  For instance, if someone picks the number 3 then I start counting from the top of the deck and say (with hand movements in brackets) "One [put it under], two [put it under], three [turn it over, put it on top FACE UP and stop].  This [show face-up card to everyone] is not your card. [Put it back face-up on top]  One [now put it under], two [put it under], three [turn over and put on top FACE UP and stop].  This etc. etc." 
Connell advises telling people to pick up a number from 1 to 7 but not allow 1.  In practice there's no need to tell people not to pick 1.  They never do (it's never happened to me). They don't pick 7 either.  And if they did pick 1, well, just turn over the top card and you're done!  Again, that never really happens.
A: The coffee mug trick is also called the Philipine Wine Trick and should be related to the Dirac String Trick, which you can find by a web search, for example here and also in my presentation Out of Line, where rotations in 3-space are related to the Projective Plane.  
A knot trick, I am not sure you would call it magic, has been shown to children and academics in many places. It requires a pentoil knot of width 20" (say 100cm)  made of copper tubing, about 0.25" (7mm)  diameter (made by a university workshop) shown in the following diagram, but without the arrows and labels:

It also needs some nice flexible boating rope. The rope is wrapped round the $x,y$ pieces according to the rule 
$$R=xyxyxy^{-1}x^{-1}y^{-1}x^{-1}y^{-1} $$
and the ends tied together, as in the following picture:

A member of the audience is then invited to come up and manipulate the loop of rope off the knot, starting by turning it upside down. This justifies the rule $R=1$. Of course the rule is the relation for the fundamental group of complement of the pentoil, which can, for the right audience, be deduced from the relations at each crossing given by the diagram 

and can be easily demonstrated with the knot and rope. (The picture has no base point and so  seems to me related more to the notion of fundamental groupoid  than fundamental group.)
It is also of interest to have a copper trefoil rather than pentoil  around to compare the relations. One warning: the use of rope does not really model the fundamental group or groupoid, so be careful with a demo for the figure eight knot! 
I did the demo for one teenager and he said:"Where did you get that formula?" This demo knot has been well travelled, for many different types of audience; on one occasion the airline lost my luggage with the rope and  I had to ask the taxi from the airport to to stop at a hardware store for me to buy d=some clothesline. I devised this trick for an undergraduate course in knot theory in the late 1970s. 
You can also see a Knot Exhibition whose aim is to use the notion of knot to explain some basic methods of mathematics to the general public.  
A: The "casting out nines" sanity check of calculations is dead simple to use (a small child can do it), but the proof requires a deeper knowledge of mathematics (more precisely of arithmetic ; my own students don't have access to it even though they know what series are and can diagonalize matrices!).
A: Matt Baker's book The Buena Vista Shuffle Club might tick the boxes. He is a math professor and card magician and the book uses a lot of math principles. It's published by Vanishing Inc. 
A: Lay out 21 cards face up in three vertical lines. Have a friend pick out any card without telling you which card he/she has chosen. Have your friend tell you which line of cards the selected card is in, and make three stacks of cards, each stack being made from each line of cards. stack the three stacks on top of each other, placing the stack with the selected card between the other two stacks (IMPORTANT!).  lay out the cards again in the exact same set up (3 lines of 7 all face up) but here is the trick: when laying out the cards, flip them face up in a line every time. In other words, don't make one line at a time, but put a card in every line one at a time.  Have your friend again tell you which line has the selected card. Stack the cards again, the exact same way you did the first time.  One more time, lay out the cards the exact same way as the last time, one card per line, and again have your friend tell you which line has the selected card. Stack all the cards again one last time, again placing the line with the selected card between the other stacked cards.  now lay out all the cards face down, one at a time. while you're doing this, remember to count, because the 11th card you place down is the selected card.  from this point you can do whatever you can think of to make the trick "magical" and shock your friend by suddenly coming up with his/her card.
A: Start with a deck of 32 cards. Then the player should take a card and tell a number $n$ between 1 and 32 then you divide the stack in 2 smaller stacks and the player has to tell which of the stacks contains his chosen card. according to a rule dependend on that number you put that stack above or below the other stack. 
After repeating this 5 times the chosen card should be exactly at position $n$. The rule has to depend on the way you want to deal cards (whether you turn around the deck and start dealing from the bottom, or you deal from the top and turn each single card around or you deal at first and then turn bost stacks around).
In one of the cases the rule was take $N-11$, find the representation in the system with base $-2$ and revert that presentation. ($0$ tells you to put the stack containing the chosen card on top, etc.).
I dont remeber this trick properly, it should not be too difficult to express the final position depending on the choices in some formula; but it is the only situation I know, in which the $-2$-system is useful.
A: i was about the to post another trick based on binary encoding in radices (or hamming codes) which has already been posted (see comment as well)
So i will post another one based on cyclic orderings (known also as "The Fitch Cheney Five-Card Trick")
excerpt (from here):

You hand a deck of cards to an audience and tell them to choose any
  five cards they wish. You collect the ve cards, look at them quickly,
  and then ask a volunteer to hide one of the cards after showing it to
  the audience. You place the remaining four cards face up in a line.
  You then tell someone in the room to go to the door and fetch your
  partner, who has been waiting outside. You take a seat in the back of
  the room somewhere out of sight. Your partner enters, takes a look at
  the four cards displayed, and correctly calls out the hidden card!
  Applause follows.

A: Requirements for audience: pen, paper, and calculator. You have nothing.
Offer the following instructions:


*

*Write down a whole number of any size without showing you.

*Just below it, write the same digits in a mixed up order.

*Subtract the smaller from the larger, and write down the result.

*In the resulting number, circle any digit which is not a 0.

*Ask them to tell you the other digits in the number which are not circled.

*You tell them the circled digit.


The key of course is that the resulting number in step 3 must be a multiple of 9 (obvious to readers here.) As the audience is reciting the digits in step 5, just keep track of the sum modulo 9. Take the result and subtract from 9 and that is the circled digit (except when the result is 0 then the circled digit is 9.)
A: How about the "Flash Mind Reader"
