Intro to Math Final Exam
The number of Hamilton Circuits in K11 is
(11x10)/2
If F34 = A and F37 = B, then F35 is
(B-A)/2
The Fibonacci numbers appear (under ideal assumptions) in
- the generations of rabbits - the generations of bees - the rings on a pineapple
The approximate value of 1/Φ is
0.618
The approximate value of F1000/F1001 is
0.618
The reciprocal of the Golden Ratio is approximately equal to
0.618
F2001/F2000 is approximately
1.618
The approximate value of Φ is
1.618
100!/99! =
100
If x is the length of a Golden Rectangle which is 1000 feet wide, then x is approximately
1618 feet.
Flatland was written in
1884
F6 + F7 =
21
The number of Hamilton circuits in the complete graph with 5 vertices is
24
In the following question(s), Fn represents the Nth Fibonacci number. F9 =
34
In the following question(s), Fn represents the Nth Fibonacci number. If F500 = a and F501 = b, then F505 = 2a + 3b None of these 3a + 5b a + 2b a + b
3a + 5b
In the following question(s), Fn represents the Nth Fibonacci number. F12/F4 =
48
50!/49! =
50
The number of edges in K11 is
55
6! =
6 x 5!
If X is the width of a Golden Rectangle whose length is 1000 feet, then the approximate value of X is
618 feet
The number of edges in K12 is
66
F6 =
8
10!/8! =
90
A, D, C, B, A
A garbage truck must pick up garbage at 4 different dump sites (A, B, C, and D) as shown in the graph below, starting and ending at A. The numbers on the edges represent distances (in miles) between locations. The truck driver wants to minimize the total length of the trip. An optimal solution to this problem is given by
A, C, D, B, E, A
A mail truck must deliver packages to 5 different store locations (A, B, C, D, and E). The trip must start and end at A. The graph below shows the distances (in miles) between locations. We want to minimize the total distance traveled. The nearest-neighbor algorithm applied to the graph yields the following solution:
Suppose a line of length X is cut into two pieces using a Golden Cut such that the longer piece has length A and the shorter piece has length B. The ratio of X to A is the same as the ratio of
A to B
Memphis, Atlanta, Houston, Dallas, Kansas City, Denver, Memphis
A traveling saleswoman's territory consists of the 6 cities shown on the following mileage chart. The saleswoman must organize a round trip that starts and ends at Memphis (her hometown) and will pass through each of the other 5 cities exactly once. The nearest-neighbor algorithm applied to this problem yields the following solution:
A, B, D, C, A
A truck must drop off furniture at 4 different homes (A, B, C, and D) as shown in the graph below, starting and ending at A. The numbers on the edges represent distances (in miles) between locations. The truck driver wants to minimize the total length of the trip. The nearest-neighbor algorithm applied to the graph yields the following solution
A, D, C, B, A
A truck must drop off furniture at 4 different homes (A, B, C, and D) as shown in the graph below, starting and ending at A. The numbers on the edges represent distances (in miles) between locations. The truck driver wants to minimize the total length of the trip. The repetitive nearest-neighbor algorithm applied to the graph yields the following solution:
regardless of the value of x
Applying the nearest neighbor algorithm with starting vertex A yields the solution A, D, C, B, A
The name of the author of Flatland is
Edwin A. Abbott
72
If the rectangle below is a golden rectangle, then the approximate value of x is
The number of Hamilton circuits in K11 is
NOT (11x10)/2
According to Binet's formula, FN =
NOT this answer
The Golden Ratio is called Phi because
Phi is the first letter in the name of the Greek architect who designed the Parthenon
has a Hamilton path that starts at G and ends at E
The following graph
has several circuits, all of which contain the edge AD
The following graph
If a line of length 1 is divided into two pieces by a Golden Cut such that the longer piece has length X, then
X is the reciprocal of the Golden Ratio
If F500 = a and F501 = b then F502 =
a + b
A Hamilton Circuit is
a circuit that travels through each vertex exactly once.
In a complete weighted graph, an optimal circuit is
a circuit with minimum total weight.
A Hamilton path is
a path that visits every vertex exactly once
The most common shape of a house in Flatland is
a pentagon
Flatland was written by
a social reformer
If F500 = a and F501 = b then F503 =
a+2b
The repetitive nearest-neighbor algorithm for solving the Traveling Salesman Problem is
an approximate and efficient algorithm
It is likely that Golden Rectangles and the Golden Ratio
appear in modern architectural designs such the United Nations General Assembly building
The nearest-neighbor algorithm is
approximate and efficient
The repetitive nearest-neighbor algorithm is
approximate and efficient.
Fibonacci Rectangles are
are approximately Golden Rectangles
Examples of Fibonacci numbers in nature
are the subject of many websites
Fibonacci first obtained the Fibonacci sequence of numbers
as the solution of a rabbit-breeding puzzle
If F100 = a and F102 = b then F101 =
b-a
If F500 = a and F501 = b then F499 =
b-a
In the following question(s), Fn represents the Nth Fibonacci number. If F500 = a and F501 = b, then F499 is 2a-b a+b None of these b-a a+2a
b-a
The Brute-Force Algorithm finds the cheapest route by
checking every possible Hamilton circuit
The nearest-neighbor algorithm
constructs a single Hamilton circuit which may or may not be optimal.
Some people suggest that the human body
contains examples of Fibonacci numbers
The Golden Ratio and its reciprocal
differ by 1
In a complete graph
each pair of vertices is connected by exactly one edge
One reason it is difficult to find a "four-leaf clover" may be that
four is not a Fibonacci number
The number of petals on many common species of flower is
frequently a Fibonacci number
Binet's formula
gives each Fibonacci number independently as the result of a direct calculation.
A complete graph
has a Hamilton circuit.
In a traveling-salesman problem we are seeking a Hamilton circuit which
has the minimum cost
In a complete weighted graph, a Hamilton circuit and its mirror-image
have the same total weight
Irregular figures in Flatland are considered to be
immoral and perpetrators of mischief
A weighted graph
is a graph whose edges have costs associated with using them to travel between vertices.
The appearance of Fibonacci numbers in patterns in nature
is an example of order and design in creation
The number of spirals on the head of a cauliflower
is often a Fibonacci number
Phyllotaxis
is the study of how plants grow
Soldiers and the lowest class of workers in Flatland are
isosceles triangles
If a graph has a Hamilton circuit then
it may or may not have an Euler circuit
The number of Hamilton circuits in the complete graph with 100 vertices is
more than a billion
Whether or not a particular graph has a Hamilton circuit
must be decided on a case by case basis by examining the graph
Traveling-salesman problems
occur in a variety of real-life contexts.
It is quite possible that the Golden Ratio occurs in many paintings and architectural designs
only approximately, because the designer seeks a pleasing proportion without specifically intending the exact Golden Ratio.
The Brute-Force Algorithm is
optimal and inefficient.
In Flatland, the author uses the unusual device of
placing the main story in a world of only two dimensions
Fibonacci
probably did NOT realize that the Fibonacci numbers occurred in many different examples in nature
Examination of the lengths involved in constructing the Parthenon suggest that the designer
purposely used the Golden Ratio in his design
An examination of some of the paintings of Leonardo da Vinci suggest that he
purposely used the Golden Ratio in the design of the painting
Flatland belongs to a literary genre known as
satire
According to the lecture material, we must be careful not to
see the Golden Ratio everywhere, since in every complex shape, some lengths will appear to have the Golden Ratio proportion by chance
Claims that the Golden Ratio appears in human art and architecture
should be carefully examined before being accepted
Trees
sometimes exhibit Fibonacci numbers in the pattern of the branches
An efficient strategy for solving a traveling-salesman problem is a method
such that the time to implement the strategy increases slowly as the graph gets larger.
Every Fibonacci number is the sum of the two previous Fibonacci numbers in the sequence, except
the first two Fibonacci numbers
Fibonacci's most important contribution to Western civilization was
the introduction of the Hindu-Arabic place-value system of numbers
(1+√5)/2 is the exact value of
the number Phi
The mirror-image of a Hamilton circuit is
the same sequence of vertices traveled in reverse order
The Golden Ratio is the only positive number such that
the square of the Golden Ratio is equal to the Golden Ratio plus 1
In a complete graph having 3 vertices
there are two Hamilton circuits.
Flatland, the novel, is broken up into
two parts of approximately equal length
The repetitive nearest-neighbor algorithm
uses each vertex in the graph as the starting point for the nearest-neighbor algorithm.
Every graph has
vertices
If we change the beginning/ending point of a Hamilton circuit then`
we get the same circuit
The appearance of the Fibonacci numbers in nature
well-documented with many examples
The solution set of the quadratic equation x2 + 8x + 7 = 0 is
x = -7 or x = -1
