1. finance
2. personal finance
3. lending
4. credit cards

MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
from FS2009
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
L ECTURE 26
Part/ References
Topic/Sections
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
T HE B EST C ARD T RICK
Notes/Speaker
Lecture 26:
The Best Card Trick
Contents
7
8
9
Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due
Multivariable GFs
19
Analytic
26.1
HowIV.3,
toIV.4
Perform
theMethods
Best CardComplex
TrickAnalysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
IV.5 V.1
26.2 Exercises
. .
Nov 2
10
11
VI.1
12
A.3/ C
Singularity Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Asymptotic methods
Asst #2 Due
1
4
Sophie
Introduction to Prob.
Mariolys
18
Limit
Laws and Comb
You
areIX.1about to pull off the most
amazing
card Marni
trick... The audience draws 5 cards from the deck
and
gives
them
to
the
volunteer
who,
in
turn,
shows
you four of the five cards, say:
Random Structures
20
IX.2
Discrete Limit Laws
Sophie
23
12
13
9
FS: Part B: IV, V, VI
Appendix B4
. . . . . . .
Stanley 99: Ch. 6
Handout #1
(self-study)
IX.3
and Limit Laws
FS: Part C
(rotating
presentations)
Combinatorial
Mariolys
4♠, K♦,
4♥, 5♣.
instances of discrete
25
Continuous Limit Laws Marni
After
aIX.4
dramatic moment of “reading”
the volunteers mind, and the minds of the audience members,
Quasi-Powers
and was hidden from you, must be the 9♠. The audience
you
reveal
that
the
fifth
card,
the
one
that
30
IX.5
Sophie
Gaussian limit laws
gasps... How could this be?1
14
Dec 10
Presentations
Asst #3 Due
How to Perform the Best Card Trick
26.1
You could come up with some story about how the first four cards were shown to you in order to allow you
to tune into your volunteers mind, thus allowing you to read the fifth card. Or explain that you are having
trouble reading his/her mind so you need the entire audience to concentrate on the fifth card. Of course all
this is all just for show. The fact is, your volunteer is actually your accomplice and he/she passed you enough
information to determine the final card. There is no element of chance here, your accomplice simply “encoded”
theMarni
value
of the
last ofcard
in the
firstFRASER
four UNIVERSITY
cards. This begs the question: How did they do this?
Dr.
MISHNA,
Department
Mathematics,
SIMON
Version of: 11-Dec-09
Let’s first describe the standard deck of playing cards. There are four suits: clubs ♣, diamonds ♦, hearts
♥, and spades ♠. Within each suit there are 13 ranks: (A)ce, 2, 3, 4, 5, 6, 7, 8, 9, 10, (J)ack, (Q)ueen, (K)ing.
It will be convenient for us to think of Ace = 1, Jack = 11, Queen = 12, and King = 13. In card jargon we are
considering “Aces to be low”, by which we mean the Ace is the lowest ranking card. There are 4 · 13 = 52 cards
in all. See Figure 1 for some examples.
(a) 5
Hearts.
of
(b) Jack of
clubs.
(c) Queen
of
Diamonds.
(d) King of
Spades.
Figure 1: Examples of names of playing cards.
1 See
original article by Michael Kleber in Mathematical Intelligencer 24 #1, 2002
Jamie Mulholland, Spring 2011
Math 302
26-1
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
Part/ References
MATH 895-4 Fall 2010
Course Schedule
L ECTURE 26
Topic/Sections
T HE B EST C ARD T RICK
Notes/Speaker
FS2009necessary for the trick? Given a collection of five cards from the deck there must be two
Why are fivefrom
cards
of 1the Sept
same
suit
(this is known in mathematics
as the pigeonhole principle). Let A and B be two cards of the
7 I.1,
I.2, I.3
Symbolic methods
Combinatorial
same suit. One of these
cards will be taken as the fifth card, i.e. the hidden card. The other card will be shown
Structures
2
14
I.4, I.5, I.6
Unlabelled structures
FS: Partthe
A.1, A.2
first, thus communicating
suit of the hidden card. All that remains now is to communicate the rank of the
Comtet74
3
21
Labelled structures I
hidden
card.II.1, II.2, II.3 Handout #1
(self study)
4
28
II.4, II.5, II.6
Labelled structures II
7
19
IV.3, IV.4
9
VI.1
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
30
IX.5
Quasi-Powers and
Gaussian limit laws
Sophie
Once the suit is known there are 12 possibilities for the rank (since the hidden card is certainly not the one
Combinatorial
Combinatorial
that
to
you). How can
the remainingAsst
3 #1
cards
5 was
Oct 5 just
III.1,shown
III.2
Due be arranged to communicate the rank? Since
parameters
Parameters
FS A.III
there are 6 ways to arrange
3 objects we can communicate a number from 1 to 6 with the remaining 3 cards.
6
12
IV.1, IV.2
Multivariable GFs
(self-study)
This doesn’t seem to be
enough information.
Complex Analysis
Analytic Methods
Does it matter which of
two
A and B, of the same suit, is shown and which is hidden? Yes, it does (to
FS:the
Part B:
IV, V,cards,
VI
8
26
Singularity Analysis
B4
do the trick IV.5
in V.1
the wayAppendix
we describe).
Let’s
define the distance, dist, between two cards of the same suit using
Stanley 99: Ch. 6
Nov 2
Asst #2 Due
the9 following
diagram:Handout #1
Asymptotic methods
10
11
12
13
(self-study)
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Sophie
Figure
2: The distance between
card ranks is the number
of steps it takes to move from one rank to the next in the
14
Dec 10
Presentations
Asst #3 Due
clockwise direction.
For example, dist(3, 6) = 3, dist(8, J) = 3, dist(Q, 5) = 6, dist(A, 8) = 7. Notice that dist(6, 3) = 10, which is not
the same as dist(3, 6) = 3. So the distance function is not symmetric in its arguments. However, we do have
dist(a, b) = 13 − dist(b, a), for any two ranks a, b. It follows that for any two ranks a, b (i.e. two cards of the
same suit) that either dist(a, b) or dist(b, a) is less than or equal to 6. If dist(a, b) ≤ 6 then we say a is smaller
than b.
So,Marni
back
to the
two cards
A, B.SIMON
We FRASER
will show
the performer the smaller
Dr.
MISHNA,
Department
of Mathematics,
UNIVERSITY
Version
of: 11-Dec-09
The larger
card (i.e. the hidden card) will be at distance at most 6 from
of the two cards and hide the larger.
the smaller card and we can use the
remaining 3 cards to communicate this distance.
All that remains now is to assign numbers 1 through 6 to arrangements of three objects. We will use the
playing cards natural ordering (from smallest to largest):
A♣ < A♦ < A♥ < A♠ < 2♣ < 2♦ < 2♥ < 2♠ < 3♣ < 3♦ < 3♥ < 3♠ < 4♣ < 4♦ < 4♥ < 4♠ < . . .
. . . < 10♣ < 10♦ < 10♥ < 10♠ < J♣ < J♦ < J♥ < J♠ < Q♣ < Q♦ < Q♥ < Q♠ < K♣ < K♦ < K♥ < K♠.
That is, cards are first ordered by rank: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, and ties are broken using the suit
ordering: ♣ < ♦ < ♥ < ♠ (i.e. alphabetic in first letter of name: (C)lub, (D)iamond, (H)eart, (S)pade). Given
three cards, say a, b, c, where a < b < c, we can list the six arrangements lexicographically (dictionary order)
and assign numbers as follows:
abc = 1
acb = 2
bac = 3
bca = 4
cab = 5
cba = 6.
Jamie Mulholland, Spring 2011
Math 302
26-2
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
Part/ References
L ECTURE 26
Topic/Sections
T HE B EST C ARD T RICK
Notes/Speaker
FS2009
For example,from
the
three cards 4♦, 5♣, K♦, have the following arrangements numbers (since 4♥ < 5♣ < K♦).
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
Combinatorial
Structures
4♥ 5♣ K♦
FS: Part A.1, A.2
5♣
Comtet74 K♦ 4♥
Handout #1
(self study)
Symbolic methods
= 1, structures
4♥ K♦
Unlabelled
= 4,
5♠ = 2,
K♦ 4♥ 5♣ = 5,
Labelled structures I
5♠ 4♥ K♦ = 3,
K♦ 5♣ 4♥ = 6.
Labelled structures II
Since the three cards had distinct ranks we don’t need to look at the suit to break ties, since there will be no
Combinatorial
Combinatorial
5 toOct
5
III.1,
III.2 an example with ties see Exercise 2(c).
Asst #1 Due
ties
break.
For
parameters
Parameters
FS A.III
6 we
12 can IV.1,
Multivariable GFs the trick. We use the term Accomplice to refer to the one who
Now
layIV.2out the(self-study)
procedure for performing
knows
all
five
cards,
and
Performer
for
the one
attempting to guess the hidden card.
7
19
IV.3, IV.4
Complex
Analysis
8
26
Procedure:
9
Nov 2
Accomplice:
IV.5 V.1
9
VI.1
18
IX.1
23
IX.3
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Singularity Analysis
Asymptotic methods
Asst #2 Due
Sophie
10 The audience select five cards at random. Let s1 , s2 , r1 , r2 , r3 be the five cards drawn from the deck,
(1)
12
A.3/ C
Introduction to Prob.
Mariolys
where
s1 and s2 have the same suit.
Limit Laws and Comb
Marni
11 Picks two of the same suit: s , s . (Note: there could be more than two cards of the same suit, just pick
(2)
2
Random Structures1
20
IX.2
Discrete Limit Laws
Sophie
any two for s1 and
andsLimit
2 .) Laws
FS: Part C
Combinatorial
Mariolys
(rotating
of discrete
12 Picks one as the hidden card: instances
(3)
after re-labeling
if necessary assume that s1 is smaller than s2 (i.e.
presentations)
25
IX.4
Continuous
Limit
Laws
Marni
dist(s
,
s
)
≤
6).
The
hidden
card
will
then
be
s
.
1 2
2
Quasi-Powers and
13 Arrange
30
IX.5the remaining 3 cards r , r , r to correspond
Sophie to the number dist(s , s ).
(4)
1Gaussian
2 limit
3 laws
1 2
14
Dec 10
#3 Due
(5)
Reveal
cards one at a Presentations
time. Reveal card s1 first,Asst
then
reveal the remaining cards in the order found in
Step (4).
Performer:
(1) Determines the hidden card: The first card s1 gives the suit of the hidden card and a place to start
counting (namely its rank). Determine the number (between 1 and 6) to which the arrangement of the
last 3 cards corresponds and add this to the rank of s1 , thus determining the hidden card s2 .
(2) Reveals the hidden card and waits for applause.
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
The Accomplice needs to be able to think fast since they need to arrange the cards rather quickly in order to
start revealing them one at a time. The Performer, however, can stall for time by drawing large copies of the
cards on the blackboard as they are revealed. This extra time will allow the Performer to work out the number
corresponding to the arrangement of the final three cards.
Warning: If the trick is performed a few times as outlined above the audience will pick up, rather quickly,
that the first card is the same suit as the hidden one. So you may want to mix up the position of the suit card
when performing the trick. It has been suggested to play the suit card in position i (mod 4) when performing
for the ith time.
Jamie Mulholland, Spring 2011
Math 302
26-3
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
I.1, I.2, I.3
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct
(a)5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov
(b)2
26.2
Sections
Part/ References
from FS2009
Exercises
Combinatorial
L ECTURE 26
Topic/Sections
T HE B EST C ARD T RICK
Notes/Speaker
Symbolic methods
1. In each scenario Structures
below, five cards, which the audience has drawn, are given. Pick a card to hide and find
2
14
I.4, I.5, I.6
Unlabelled structures
A.2
an arrangement FS:
of Part
theA.1,
remaining
four cards which determines the hidden card.
10
11
Labelled structures I
Labelled structures II
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Complex Analysis
Asst #1 Due
Multivariable GFs
Singularity Analysis
Asymptotic methods
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
(c)
12
IV.5 V.1
Comtet74
Handout #1
(self study)
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Sophie
2. Your
accomplice
has presented you
with the
Quasi-Powers
and following arrangements of four cards. Determine the fifth
30
IX.5
Sophie
Gaussian limit laws
card. You may assume the first card
is the suit card.
13
14
Dec 10
Presentations
Asst #3 Due
(a)
(b)
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
(c)
Jamie Mulholland, Spring 2011
Math 302
26-4
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
Part/ References
from FS2009
Solutions to Exercises:
1.
(b) Hide the
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
11
12
Topic/Sections
T HE B EST C ARD T RICK
Notes/Speaker
Symbolic methods
Combinatorial
(a) Hide the 9♣ and
present the arrangement:
Structures
6
10
L ECTURE 26
9 (c)
12
Labelled structures I
Labelled structures II
IX.1
20
IX.2
23
IX.3
13
30
14
Dec 10
Asst #1 Due
Multivariable GFs
Complex Analysis
Analytic Methods
FS: Part B: IV, V, VI
Singularity Analysis
Appendix B4
Stanley 99: Ch. 6
Asymptotic methods
Handout #1
(self-study)
three
possible choices here:
(ii) hide the
IX.4
Combinatorial
Parameters
the arrangement:
There
VI.1 are
(i)A.3/
hide
the 5♦ and present:
C
18
25
Combinatorial
parameters
A.IIIpresent
K♥ FS
and
(self-study)
IV.5 V.1
Unlabelled structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
J♦
and present:
IX.5
Asst #2 Due
Sophie
Introduction to Prob.
Mariolys
Limit Laws and Comb
Marni
Discrete Limit Laws
Sophie
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Sophie
Presentations
Asst #3 Due
(i) hide the 3♦ and present:
In this last case we need to use the suit ordering to break the tie between the two 5’s: recall ♦ < ♠.
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
2. (a)
(b)
Jamie Mulholland, Spring 2011
Math 302
(c)
26-5