Dunno if this counts, but somewhere around when we learn the pythagorean formulas, I usually let the kids tell me the square of a two digit number with a calculator, and wow them with a nearly instantaneous revelation of their original number.

Ex: They say 1369, and I say 37.

The birthday guessing trick is a wow-producer.

One example is:
http://www.teach-nology.com/worksheets/math/magic/bday/

I used [a variation of] the Three Swap with my students, eventually moving to an Excel spreadsheet (recreated here in Google Drive) to try really big or really small or negative numbers. For the visual students, a desmos graph shows how the end result doesn’t change, even when their number does. Then I had them design their own to show their families. A few came back the next day, proud of their “magic”.

The Magical Triangle Theorem served some of this purpose, though that was more for the sake of drama (which earned me some points that day).

I use an ESP activity on the day I plan to introduce composite functions; it’s similar to your “number trick” example above. Blog post with video explanation – http://mathcoachblog.com/2012/10/18/composite-functions-and-e-s-p/

Also, this card trick I did recently in class hooked many kids, and leaving them wanted to know how it worked. There a fun discussion of transformations to be had – that two cards dealt face down is equivalent to two cards swapped in my hand and then dealt: http://mathcoachblog.com/2014/09/24/class-opener-day-17-time-for-a-card-trick/

Modular arithmetic trick: Give me a perfect cube between 0 and a million, and I’ll immediately tell you its cube root.

If you’re much better at listening you can actually go up to a billion, but it’s pretty hard. Art Benjamin has the details: https://www.math.hmc.edu/~benjamin/papers/Cubing.pdf

For more combinatorial classes I love the Fitch Cheney card trick, which requires two performers who know what’s going on. Audience member hands me five cards. I lay four of them out on a table in some order, then hand one card back to the audience member (who olds onto it). Assistant enters the room, looks at the four cards, and says what the fifth one was.

How is this possible? There are only 24 ways to arrange four cards (and I’m not using any upside-down trickery), but there are 48 other cards in the deck that could be the answer, so it seems like there’s not enough information. Good writeup here: http://www5.spelman.edu/~colm/fitch.pdf

Art Benjamin has a lot of cool tricks. In the mathemagician ted talk he squares large numbers, with some practice anybody can do any two digit numbers using some algebra of perfect squares.
(a+b)^2= a^2+2ab+b^2

Example: 53^2 is (50+3)^2=50^2 + 2(50)(3) + 3^2
The quick way to do that in your head is square the first digit 25 put a 0, if necessary, then square the second digit 9, 2509 then double one of the digits and multiply it by the other and add it to your middle two digits 5*2=10 ->*3=30 2509+300=2809 (or multiply and double, your choice)

Try 84^2

(64)(16)+(6)(4)=7056

I’ve done this one with my algebra students leading into the perfect square trinomial.

Here is one I really liked that involved student’s phone numbers.

http://themathbehindthemagic.wordpress.com/2013/10/07/the-math-behind-the-magic/

Nrich has some lovely magic with cards:
http://nrich.maths.org/public/search.php?search=card+trick
Also some nice mathemagic number tricks: http://nrich.maths.org/1051

A nice trick is this one with dice. A lot of dice. Let’s say 50 or so. You lay them on the ground like a long chain. The upward facing numbers should be completely random. Then you go from the one end to the other following the following rule. Look at the number of the die where you’re at. Take that many steps along the chain, towards the other end. Repeat. If you’re lucky, you already end up exactly at the last die. You’ll be a magician immediately! But usually, that isn’t the case. What you usually have to do, is take away all those dice which you jumped over during the last step. Tell them that that is “the rule during the first round”. Now the actual magic begins. You tell the audience that they can do whatever they want with the first half of the chain. They may turn around dice. Swap dice. Take dice away. Whatever. As long as they don’t do anything with the second half of the chain. [If you like risks, let them mess up a larger part of the chain.] What you’ll see, is that each and every time, they will end up exactly at the end of the chain!

I made a video of this, but I did it in Dutch, so the bonus is that you learn how to count in Dutch: een, twee, drie, vier, vijf, zes, zeven, acht, negen, tien ;-)

One of my favorite games to play with students is “31”. You start by letting students know that we will alternate turns, each subtracting a number ranging from 1-X. The person who is left with 1 is the looser.

I typically play this game to begin with for something like a “free homework pass.” As you can guess, this is modular arithmetic where you need to get to a multiple of (X+1) +1. When you first start out, you can randomly subtract things as a teacher to try to stimulate different student thinking. I typically pick the first interval to be 1-5 so “key numbers” of the game are 1,7,13,19,25 and 31. I always ask the student if they want to go first of if they want me to.

It typically takes a few times playing the game before students really catch on, but it is great to see them create ideas and want to test their theory against the teacher. It is also great to see them realize their theory is wrong, and try to create a new one while playing the game. I typically get numerous requests to play the game during the year, and I will let their thinking cool down for a month or two before accepting the challenge again- but I normally throw in the twist that I pick the student to play against. It keeps them all on their toes and accountable for remembering the mathematics behind the game.

A few years ago, I found this “trick” on a “maths” site, not sure which, but it was UK. You need 5 index cards. Number them 1, 2, 3, 4, 5 in red ink on the front. On the reverse side, number them 6, 7, 8, 9, 10 in blue ink. Be sure that 1 and 6 are on opposite sides of the same card…same with 2 and 7, etc. Turn your back to the group of students. Have one of the students drop the 5 cards on the floor and tell you how many cards landed with the blue number face up (they don’t tell you the number, just “3 cards are written in blue”). Tell them the total of the numbers showing is 30. The key is that each blue number is 5 more than its respective red number. Red numbers total 15. Each blue number raises the total by 5. So 3 blue numbers make it 15 (the basic sum) + 15 (3 times 5). Let them figure out how you are using the number of blue numbers to find the total of the exposed numbers.