Mathematics for Computer Science
MIT 6.042J/18.062J
Tricks with Counting
& Matching
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.1
The Magic Trick
• Audience chooses 5 cards
• Assistant reveals 4 of them
th
• Magician announces 5 card
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.2
The Assistant’s Choices
• Decide the order of the 4 cards
4! = 24 orderings
-- but 48 cards remain
• Decide which 4 cards to reveal
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.3
Match 5-Card Hands with
4-Card Permutations
5-card hands
(no order)
?
4-card perms
(ordered)
Hand must match
to cards in it
Which one to pick?
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.4
Match 5-Card Hands
with 4-Card Permutations
5-card hands
(no order)
?
4-card perms
(ordered)
How can we ensure
consistency?
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.5
Matchings
Women
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
Men
lec 10m.6
Matching for the Women
Women
Men
women all have different husbands
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.7
Neighbor Sets
For subset S of women,
N(S) is set of neighboring men
S
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
N(S)
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Hall’s Marriage Theorem
Matching if and only if
|S|  |N(S)|
for all S.
|S|=3
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
|N(S)| = 5
lec 10m.9
Match 5-Card Hands
with 4-Card Permutations
Lemma: If woman degree is c,
and man degree is d,
then match for the women iff
c≥d
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.10
Match 5-Card Hands
with 4-Card Permutations
5
deg     4!  120
 4
deg  52  4  48
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.11
Match 5-Card Hands
with 4-Card Permutations
So there is a match the
Magician and Assistant can use.
In fact, there is one even for a
double-size deck.
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.12
A Memorable Matching?
 52 
matches
of
hands
   2,598,960
to sequences
5 
How will Magician & Assistant
learn them?
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.13
Magic Trick Revealed (I)
Among 5 cards chosen:
at least 2 have same suit
(Pigeonhole Principle)
Assistant uses one of them 1st
Aha! The first card has the
same suit as the hidden card!
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.14
Magic Trick Revealed (II)
How do we figure out the
value of the hidden card?
Aha! Look at the order
of the other 3 cards!
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.15
Magic Trick Revealed (II)
Fix ordering of the deck
A♣ < 2♣ < 3♣ < … < K♣ <
A♦ < 2♦ < 3♦ < … < K♦ <
A♥ < 2♥ < 3♥ < … < K♥ <
A♠ < 2♠ < 3♠ < … < K♠
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.16
Magic Trick Revealed (II)
• Eliminating the value of the 1st card
leaves 12 values for the hidden card.
• Possible orders for the next 3 cards:
{ SML, SLM, MSL, MLS, LSM, LMS }
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.17
Magic Trick Revealed (II)
Wait! Only have 6 permutations for the
next 3 revealed cards.
Hidden card has 12 possible values!
Of the two cards with the same suit,
the choice of which is revealed can
give 1 more bit of information! Aha!
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.18
Clockwise Distance
Between any 2 of the 13 card values, the
smaller clockwise distance is at most 6:
Q K A
Hide the
card with
the smaller
offset.
J
10
7
9
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
2
3
4
6
8
7
6
5
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Magic Trick Revealed (Finally)
• The first card determines the suit
(♠ ♥ ♦ ♣) of the hidden card.
• Hidden-card value (A…K)
= first-card value + offset (≤ 6).
• The offset is determined by the
permutation of the other three cards:
SML = 1, SLM = 2, MSL = 3,
MLS = 4, LSM = 5, LMS = 6.
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.20
Example
Hidden:
First:
Offset = 1 = SML:
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.21
Why the Magic Trick Cannot Work
with Only 4 Cards:
Audience can
pick any 4-card
hand:
Assistant can
reveal a 3-card
sequence:
 52 
   270, 725
4
52!
 132, 600
49!
Pigeonhole: 3 hands map to same sequence
Magician won’t know which one!
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.22
Team Problems
Problems
1--4
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.23
Magic Trick Revealed
• The first card determines the suit
(♠ ♥ ♦ ♣) of the hidden card.
• Hidden-card value
= first-card value + offset (≤ 6).
• The offset is determined by the
permutation of the other three cards:
SML = 1, SLM = 2, MSL = 3,
MLS = 4, LSM = 5, LMS = 6.
Copyright © 2005 by Albert R. Meyer. All rights reserved. November 7, 2005.
lec 10m.24