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Sheng Liu, Zhixia Lin, Tianyu Chen, Renteng Huang
Professor Chi-Kwong Li
A114
22 June 2014
Mathematical Magic
Abstract
Magic is a performing art that entertains audiences by staging tricks or creating illusions
of seemingly impossible or supernatural feats using natural means. But neither of them is
paranormal. The methodology behind magic is often referred to a science whilst the
performance aspect is more of an art form. Mathematics is one of the subjects that can relate
to magic. Here are some fairly interesting magic and mathematics. Each of them both have a
lovely bit of magic and a genius bit of math behind it.
1. 27 cards trick
Routine:
Step 1: choose one volunteer.
Step 2: let him/her choose a card and do not let the magician know about the card.
Step 3: put the card back and mix the cards.
Step 4: the volunteer choose one number between 1 and 27.
Step 5: divide the cards into 3 piles and after dividing, the volunteer point out which pile
the card is in.
Step 6: put the 3 piles together and repeat Step 5 for another two times.
Step 7: show the appointed card.
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Explanation:
We start with 27 cards, and the volunteer pick one card, remember it and put it back
randomly. The magician don’t know where it is. After the first round, the magician can ask
which pile the card is in. Then the magician can choose to put the special pile on the top,
middle or bottom.
Since there are three rounds, the magician has three times to choose, every time there are
three options. It makes 27 different situations which lead to different final position of the
chosen card. So it’s no coincidence in this trick.
Without loss of generality, we assume the card is the a th, counting from top to bottom.
After putting them into 3 piles, the card must be the [a/3] th card of the (a mod 3) pile.
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
Pile 1
1
4
7
10
13
16
19
22
25
Pile 2
2
5
8
11
14
17
20
23
26
Pile 3
3
6
9
12
15
18
21
24
27
After the first time we picking them up, we assume the card is the b th, counting from
top to bottom. The follow the formula as following:
After the 1st round, a th->b th:
After the 2nd round, b th->c th:
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For example, if we choose middle, bottom, middle.
As 27 is the cube number of 3, so [[[𝑎/3]/3]/3] = 1. So the final position is 17.
If we choose bottom top bottom, the final position is 21.
We can see that the original position has nothing to do with the final position. It all
depends on how we combine different options.
In fact, we could use base 3 numbers or which we called ternary, to calculate the
combination, and we could put the card into any given situation.
1
2
3
4
5
…
24
25
26
27
000
001
002
010
011
…
212
220
221
222
After transfusion, 0 stands for top, 1 stands for middle and 2 stands for bottom. And it
should be read in reverse (e.g. 221 means middle, bottom, bottom).
Conclusion: We can use base 3 numbers or what is called ternary to calculate what
combination of top middle and bottom will put the card into any given position .The
mathematical method for moving cards to a selected position in a pack can also be used to
efficiently move data in computer memory.
2. Talking jokers.
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Routine:
Step 1: two jokers are put in one pack face up in random, other cards are face down. Ask
two volunteers to help.
Step 2: one volunteer takes roughly one-third of the pack and the other takes half of the
rest. Ask them to remember the last card of the pile in their hands.
Step 3: take back the two packs, the two jokers will tell us where the two cards are.
Explanation:
Figure 1
Figure 2
Figure 1 shows the front view of a deck of poker, the first red line represent for Joker 1
and the second line represent for Joker 2.
Step 1. Before we play the trick, we put two jokers in and remember the position of
them, respectively noted by x and y. Hence, x and y are the numbers we have already knew.
Step 2. We split the cards into 3 parts, as we can see from Figure 2, using yellow, blue,
and red to show it, and we assume that m cards in top part(yellow part), and n cards in middle
part(blue part).
Step 3. Show audience the m th card and n th card, and ask them to remember two cards
(including suit and face value)
So OUR JOB now is to solve the problem below:
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given: x, y
Find: m, n;
Step 4. (Core) We put 3 parts together, BUT this time we change the order so that we can
locate Joker 1 and Joker 2 and find m and n. As we can see from figure 3.
Figure 3
Figure 4
In fact, we exchange the order of first and second part, so the location of Joker 2 and
Joker 1 turns into y-m and x+n respectively.
Step 5. counting out the number of y-m and x+n when we pretend to find Joker 1 and
Joker 2, so we solve the problem above and find out m and n.
Step 6. Last thing we need to do is calculate the the m+n th and n th card pick them out
by counting, we get the card we want.
So things turns out that the whole Magic thing is just a arithmetical thing. We hope you
can enjoy mathematics as same as you enjoy this little trick.
Conclusion: This trick is based on the two jokers and needs a series of calculation.
From the trick, we can know that Mathematical magic is a kind of magic which establish on
mathematical formula and calculation, such as equations, systems, and so on. In daily life,
everything has a reason. What we should do is to find the regular out and even can skillfully
use it. Just like the two jokers, we just need to remember the order of them, and list two
equations to solve the problem.
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3. The voice controlled path mystery.
http://v.163.com/movie/2012/8/A/U/M8JHGS5KF_M8JHL46AU.html
Works Cited
Online courses of Queen Mary, University of London: Mathematical Magic.
http://v.163.com/special/opencourse/magic.html
Peter McOwan with Matt Parker. “Mathematical Magic” Serial No. 1089
Magic (illusion).Wikipedia.
http://en.wikipedia.org/wiki/Magic_(illusion)#Categories_of_effects