Counting
Example:
If we draw 37 circles in the plane, such that every two circles intersect
in two points and such that no three circles intersect in the same point
("mutually overlapping circles in general position"), how many regions in
the plane do we
1
Impartial Games
An impartial game is a two-player game in which players take turns to make
moves, and where the moves available from a given position dont depend
on whose turn it is.
A player loses if they cant make a move on their turn (i.e. a player w
Multisets
Example:
A bag of Scrabble tiles contains 100 tiles: 10 A's, 2 B's, 2 C's, 5 D's and so on.
When you start a game, you take 7 letters from the bag, and put them on a
rack. How many possible hands can you get, if we say that the order of the
tile
1:Nim
Nim: finitely many piles of coins; a move comprises removing a positive number
of coins from a single pile; a player loses if they can't move.
Remark:
For any nim position P, either it can be won by the player with the move,
or it can be won by the
1
Solutions to Midterm 2
1. (a) [diagram omitted]
(b) A := cfw_8, 12, 9, 5, 7, 11 is an antichain of size 6,
and the chains
C1 := cfw_1, 2, 4, 8
C2 := cfw_3, 6, 12
C3 := cfw_5, 10
C4 := cfw_7
C5 := cfw_9
C6 := cfw_11
form a chain partition of X into 6 cha
Permutations
A _permutation_ of a finite set S is an ordered list of its elements.
An _r-permutation_ of S is an ordered list of r of its elements.
Warning:
there is another, related, meaning of 'permutation': an element of the
group of bijections of S. W
Math 3U3, Test 1
Bradd Hart, Feb. 7, 2013
Please write complete answers to all of the questions in the test booklet provided. Partial credit may be given for your work. Unless otherwise noted, you
need to justify your solutions in order to receive full cr
MATH 3U03 Winter 2015 Midterm 1
Martin Bays
Midterm 1
Duration of test: 50 minutes
McMaster University
2015
Write complete answers to 3 of the 4 questions. Partial credit may be given.
You must justify your solutions to get full marks. You may use results
Dr. Bradd Hart
Math 3U03
Apr. 12, 2013
This examination is 3 hours in length. Attempt all questions. The total number of available
points is 54. Marks are indicated next to each question.
Write your answers in the booklets
provided. You must Show your wor
MATH 3U03 Winter 2015 Midterm 2 SAMPLE
Martin Bays
Midterm 2 SAMPLE
Duration of test: 50 minutes
McMaster University
2015
Write complete answers to all questions. Partial credit may be given.
You must justify your solutions to get full marks. You may use
1
Ch. 1; Q. 29:
Consider the Nim position with piles of sizes 22,19,14,11. Is it balanced
(Nim sum 0)? Suppose the rst player moves by taking 6 coins from the pile
of size 19. How should the second player respond?
In binary, our piles are of sizes:
10110
1
Ch. 3; Q. 9:
There are 10 people in a room, with integral ages between 1 and 60 inclusive.
Prove that there are two disjoint groups of people with equal total age. Can
10 be reduced?
There are 210 = 1024 subsets of people. The total age of the room is a
1
Ch. 4; Q. 38:
Let (X1 , 1 ) and (X2 , 2 ) be posets. Dene a relation (the product order) T
on X1 X2 by (x1 , x2 )T (x1 , x2 ) i x1 1 x1 and x2 2 x2 .
Prove that (X1 X2 , T ) is a poset.
We prove this more generally for a product of m posets (Xi , i ).
W
1
Ch.7 Q.40:
Let an be the number of ternary strings of length n made up of 0s, 1s, and
2s, such that the substrings 00,01,10,and 11 never occur.
Prove that an = an1 + 2an2 for n 2, and a0 = 1, a1 = 3.
Hence nd a formula for an .
The only string of length
1
Ch.7 Q.11:
The Lucas numbers l0 , l1 , ., are dened by
ln+2 = ln + ln+1 ; l0 = 2; l1 = 1.
Prove that
(a) ln = fn1 + fn+1 for n 1;
(b)
n
2
i=0 li
= ln ln+1 + 2 for n 0.
(a) l2 = 3, so this relation holds for n = 1 and n = 2.
Let dn := ln fn1 fn+1 .
Then