Space Mystery Magic Trick

January 30th, 2013 by Math Tricks | No Comments | Filed in Math Tricks, Teaching Tricks

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A Math Magic Trick

This is a great trick just for the sake of performing great tricks.  BUT – it is also a great way for teachers and parents to make algebra interesting to kids!  For all of you teachers and parents out there looking for teaching tricks that will help kids understand algebra, this is one to put into your “teaching tool chest”!  I’ll give you the algebraic explanation of the trick below.  Here, I will tell you about the trick and explain how to do it.

You start off with a stack of “Space Mystery” cards (which you can make yourself with our free PDF – the link to that is below), half of which have aliens facing up, and the other have space-only cards facing up.

Performing the Space Mystery magic trick is pretty easy once you know how.  You start off with an even number of cards – I would suggest 6 to start off with, and then you can work your way up to whatever number you want, but I would suggest no more than 12.

Set up the set of cards such that half of them are showing the alien side, and the other half are showing the space-only side.  Place the cards on the table to demonstrate to your audience that half of the cards are showing the alien side:

Now pick up the cards and shuffle them well:

Place half of the cards on the table in a single row one-by-one:

With the remaining cards in your hand, use slight of hand to flip them.  I like to switch them into the other hand while I flip them, making it harder for your audience to detect what you did.  Now place the remaining cards on the table in a second row one-by-one.  Magically, the number of cards showing the alien side are the same in each row:

As you become more proficient with the trick, you can spiff it up by separating the alien-sided cards from the space-only cards in each row to make the illusion look more striking.

Why does this trick work?

How this math magic trick works can be best explained algebraically.  I will use the case where 12 cards are being used.  With a little patience as you go through the explanation, you will soon get the “Ah-Ha!” moment:

You start off with 6 alien cards face-up

After the cards are randomized, they are split into 2 stacks of cards, 6 cards each

Let the number of alien cards in stack 1 = A

Then the number of alien cards in stack 2 must = 6-A

Now, with a little thought you can determine the number of space-only cards that remain in stack 2.  This turns out to be 6 minus the # of alien cards, or:

6 – (6 – A), which is equal to A

So, if you now flip stack 2, the space-only cards (equal to A) now become alien cards – the same number of alien cards in stack 1!

Printable Cards

Here is the free PDF (Space Mystery Printable Cards.pdf) that you can use to print your own cards.  I used the Avery white, two-sided, clean edge business cards (Avery #28878) to print them out.  These can be purchased at any office supply store – I got mine at Walmart!  They came out pretty well – very uniform from card to card.

The Alien Storyline

If you are able to perform a dramatic presentation, here is a storyline I came up with that you can use while doing the trick.  You explain to the audience the story behind the aliens:

The Talletians are a race of highly intelligent beings from a star system 32 parsecs from our own system.  Throughout their history, they have had an obsession with symmetry.  This obsession has led them to rapid advances in science and technology, which in turn gave them an ability to quickly conquer space.

The Talletian obsession with symmetry is evident when they travel through space.  When they are in groups, they prefer to travel such that they are present in equal numbers between groups.  They do not tolerate broken symmetry.

You can witness this “need” for symmetry yourself.  The 12 cards in this pack are identical; one side displays a region of space, and the other side displays a Talletian in that space.  Start off by dividing the cards in two groups of six cards each – one with the space side up, and the other with the alien side up.  Now shuffle the cards several times to insure a random distribution.  Now divide the cards into two piles of 6 cards each.  Each pile now has the randomly distributed alien-up cards.  Break this randomness by flipping one pile over.  Now, if you inspect each card in each pile, you will see that each pile contains the same number of alien-up cards.  Symmetry has been made!

A Valuable Math Con for Fun and Profit

August 5th, 2012 by Math Tricks | No Comments | Filed in Math Tricks

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Rock dulls scissors.

Scissors cut paper.

Paper covers rock.

Rock dulls scissors.

And so on, and so on . . .

Most everybody (if not all) I know has played Rock-Paper-Scissors.  It is an interesting little competition, but have you ever given it any thought beyond its use as a simple game to be played between two people?  It does not seem to follow transitive logic.  For example:

If A is over B, and B is over C, then A is over C

is an example of a transitive relation.  In the case of the game Rock-Paper-Scissors, there is no transitive relationship; it is intransitive.  To make the difference between transitive and intransitive clear, I’ll give you another example of each.  Jack is taller than Bill, and Bill is taller than Peter.  Therefore, Jack must be taller than Peter.  This physical relationship between Jack, Bill, and Peter is transitive.  However, the relationship Jack is a friend of Bill, and Bill is a friend of Peter does not necessarily mean that Jack and Peter are friends (although they very well may be); the relationship is intransitive.

So I can hear you thinking, “That’s all very interesting, but so what?”  So what?  So what if I tell you that you can use this property to make some cash off of some unsuspecting friends – or a LOT of money off of some not so friendly!  Well, never mind what I just said there – I would not want you to engage in anything illicit like gambling.  I will tell you, however, that the relationship “is more likely than” is intransitive, and you can use this to your advantage.  In essence, it will be like knowing in advance which item your opponent will be using during a game of Rock-Paper-Scissors.

A method that you can use to cheat your friends is intransitive dice.  It is a set of three color-coded dice with non-standard numbers on the faces.  Each die has three numbers, repeated twice:

Intransitive dice (opposite sides have the same value as those shown)

Like Rock-Paper-Scissors, one die has an advantage over another.  In this case, the red die has an advantage over the green die, the green die has an advantage over the purple die, and the purple die has an advantage over the red die.

The game you play with these dice is simple.  Each player rolls a die, and the player with the higher number wins the round, and is awarded one point.  Players roll 20 times each, and the player with the most points after 20 rounds wins the game.  To play, have your opponent choose a die to roll.  Now comes the trick – if your opponent chooses green, you choose red.  If he (or she) chooses purple, you choose green.  And if your opponent goes for red, you go for purple.  Each of these scenarios gives you the advantage, namely, a 5/9 (or a 55.55% chance) of winning!

To see how the probabilities are calculated, realize that there are nine possible outcomes during each round of play – player one gets 1 of 3 possible numbers and player two counters with 1 of 3 possible numbers, giving  a total of 9 possible outcomes (I’ll rule out the possibility of dice landing on their edges).  Now look at the numbers on the red and green dice.  The possible ways for red to win would be:

9 over 8

9 over 6

9 over 1

4 over 1

2 over 1

Thus, red has 5 possible ways of winning out of 9 possible outcomes.  Look at the green vs. purple battle.  The possible ways for green to win are:

8 over 7

8 over 5

8 over 3

6 over 5

6 over 3

Again, 5 out of 9 ways to win.  Lastly, look at the ways purple can win over red:

7 over 4

7 over 2

5 over 4

5 over 2

3 over 2

giving the purple a greater than 55% chance of winning.

Pretty cool stuff, eh?  I don’t know if there are any games in casinos that utilize intransitive dice.  If you want intransitive dice, there are some available through the web.  Also, Amazon.com has blank dice available that you can put your own numbers on.  Whatever you do, have fun with it, but please don’t fleece anybody too badly!