Perspective
What is the slope and y-intercept of y = −2?
slope:0
y-intercept: -2
Follow chart 1:
What is the payoff for Player 1 of the outcome (B, X)?
What is the payoff for Player 2 of the outcome (B, X)?
What outcome is best for Player 1?
What outcome is best for Player 2?
9
2
C,X
AZ
Draw a cyclic graph that is NOT bipartite
Triangle, Pentagon, heptagon, etc (will check board)
Find the plurality and two-round runoff :
V 7 5 14 9 13
P B A A B C
C B D D B
A D C A A
D C B C D
A > B > C > D
B > A > D ∼ C
What are the symmetries of an equilateral triangle? Name them (including Trivial symmetry)
Trivial Symmetry
Reflection in line m1, m2, and m3
Rotations 1/3, 2/3, turn clockwise
Let u = (4, 2).
Compute the following vectors:
0⃗u . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1/2⃗u . . . . . . . . . . . . . . . . . . . . . . . . . . .
2⃗u . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3⃗u . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(0,0)
(2,1)
(8,4)
(12,6)
Write the sample space for a fair 12-sided die rolled once:
(1,2,3,4,5,6,7,8,9,10,11,12)
Eulerize the graph by duplicating as few edges as possible.
(will check board)
Calculate the social ranking under the combined approval voting system:
v1 v2 v3 v4 v5
A +1 -1 -1 +1 +1
B +1 -1 +1 +1 +1
C 0 0 0 +1 +1
D 0 +1 0 0 0
B>C>A~D
Draw a shape for D5 and C2
D5: pentagon (will check board)
C2: (will check board)
1. Find the parametric equation of the line passing through the points (2, 1, 2) and (3, 3, 3).
2. Find the vanishing point
1. (2, 1, 2) + t (1,2,1)
2. (1,2,1)
follow chart 2 and answer the following:
All Dominance Relations:
Strictly Dominated Strategies:
Strictly Dominant Strategies:
B>A, B>C, Z>Y, Z>X
A,C,X,Y
B,Z
look at graph 5: Apply the Greedy Coloring Algorithm to the following graph: (You should color the vertices in the order indicated, and use the ‘colors’ A, B, C, D, etc...)
Find a coloring using the minimal number of colors possible.
State the chromatic number of the graph:
(will check board)
(will check board)
3
Assume the voting system is Borda
Calculate the social ranking, assuming the voters voted sincerely.
V 20 11 12
P B C A
C A B
A D C
D B D
C > B > A > D
What is the symmetry type for the letters A, P, & Z?
D1, C1, C2
the line in the xy-plane passing through the points (3, 4) and (−2, 1)
What is its equation y = mx + b form?
y= 3/5x + 11/5
Follow chart 3: Apply the IESDS algorithm to the following game (remember to state the relevant dominance relation and to redraw the payoff matrix after each elimination):
A, X
Suppose G is a connected planar graph with 330 vertices and 650 edges. If G were drawn without any edge crossings, into how many regions would G divide the plane? In other words, how many faces does G have?
322 faces
look at chart 6:
Consider the following electoral map. There are 2 alternatives - Yellow Circles and Blue Diamonds (or grey and black in black and white printing), 25 voters, and 5 seats/districts available. Each district has 5 voters. We see that exactly 13 (i.e. 52%) voters vote Blue.
Divide the grid into 5 districts, each with 5 voters, so that Blue wins 4 of the 5 districts. (Your districts should be connected.)
will check board
Complete the composition table for the above shape on chart 7.
then answer this question:
What symmetry x makes the equation, r ◦ x ◦ m1 = I, true?
will check board
x= m4
A tree is 50 ft tall. You are standing 150 ft from the tree and you centrally project the tree onto a canvas 3 ft away. How tall is the image of the tree on the canvas?
How far do you need to be from the tree so that it appears to be 3 ft tall on the canvas?
hint: d/D= img
1. 1 ft
2. 50 ft
Follow chart 4:
Suppose Player 1 is using a mixed strategy, and they chooses strategy A with probability p.
Write formulas for the expected payoffs of each of Player 2’s strategies.
Expected payoff for strategy X, E2(X) =
Expected payoff for strategy Y , E2(Y ) =
What value of p makes Player 2’s expected payoffs the same for both of their strategies? (This is called Player 1’s indifference strategy.)
-2*p + 2(1-p) =2-4p
1*p - 2(1-p) = 3p-2
2-4p = 3p-2 ---> 7p=4 ---> p= 4/7
Answer the following for K 2,4:
v=
e=
f=
degree sequence=
chromatic #=
6
8
4
(2,2,2,2,4,4)
2
Imagine a state with 200 voters. There are two parties, party A, party B. The state is divided into four districts, each with 50 voters.
D1 D2 D3 D4
Party A 26 26 26 0
Party B 24 24 24 5
Calculate Party A’s total vote %:
Calculate Party A’s total seat %:
Calculate Party A’s total wasted votes:
Calculate Party B’s total wasted votes:
1. 39%
2. 75%
3. 0%
4. 96%
Consider the following symmetries of the plane:
• t: the translation left 2 units (i.e. the translation (1, 0))
• r: the rotation about (0, 0) by 180◦, clockwise
• mx: the reflection in the x-axis
• my: the reflection in the y-axis
Q1: Where does the symmetry (mx ◦ t) take the point (1, 0)?
Q2: Where does the symmetry r take the point (2, 1)?
1. (-1,0)
2. (-2,-1)