For Students Taking
Discrete Math Honors
Summer Review Packet
Dear Student and Parent/ Guardian,
The purpose of this summer work is to help prepare you for your upcoming discrete math honors
course. Discrete Math is the study of mathematical structures dealing primarily with “finite
processes” and “countable sets”. These ideas are fundamental to the science and technology of
the computer age. Discrete math is an elective course that is fun and especially suited for
students who are talented in math, enjoy math, want to be challenged, and/or are thinking
about careers in math, computer science, engineering, or business. Discrete math is a problem
solving course. The main themes in an introductory course in discrete math are pattern
recognition, induction and recursion, sequences and series, number theory, logic, proof and
truth tables, combinatorics, discrete probability, matrix algebra & applications, Boolean algebra
and graph theory applications and modeling.
Electing to study discrete math indicates that you, the student, have an interest in studying
advanced topics of mathematics. The math department at Plymouth Public Schools wants you
to be successful in this course. This summer work is designed to help you explore problem
solving strategies such as: drawing a diagram, making a table or list, looking for patterns, solve
a simpler problem, guess and check, the use of logical reasoning. Please be aware of the
following guidelines.
 The packet is due the first day when you return. (You will receive half credit if turned in
the second day and no credit after that.)
 Every problem must be completed. None left blank.
 The packet is worth 8 times a regular homework grade.
 Graphing calculators are allowed.
 Work or an explanation must be shown to receive credit!
 Final answers must be circled.
 When you return to school, we will discuss the various strategies and solutions.
I hope that you have an enjoyable summer and return to school ready to be successful in
Discrete Math Honors. I look forward to exploring the topics in discrete math with you in the
upcoming 2017-18 school year! Feel free to email me at [email protected] with
questions or concerns regarding this summer work.
Discrete Math Honors
Summer Work
This assignment is due the first day of school. It will count as 8 homework assignments. As stated
above, you must show work or give an explanation. Collaborating, discussing or getting help from
your peers, parents, or teachers is fine, but your work must be your own. Do not copy from a
classmate. Honesty and integrity are expected of everyone.
1. These are geometric illustrations of triangular numbers. State the next 3 triangular numbers in
this sequence.
2. . What is the next number in the following sequence?
2, 6, 30, 210, 2310, 30030, .......
3. State the next 3 numbers in each sequence.
a) 4, 7, 11, 16, 22, ……
b) 1, 5, 13, 29, 61, .…
c)
1, 4, 13, 40, 121, .…..
d)
3,
e)
5, 6, 10, 19, 35, ……
7, 15, 31, 63, ….
4.
Each edge of a cube is colored either red or black. If every face of the cube has at least one
black edge, what is the smallest possible number of black edges?
5.
Each of four students hands in a homework paper. Later the teacher hands back the graded
papers randomly, one to each of the students. In how many ways can every student receive
someone else’s paper?
6. Three horses run a race. In how many different ways can the 3 horses finish the race assuming
that ties are allowed?
7. Malcolm used to brag that he had 11 Hondas at his estate; all were either motorcycles or cars.
In all, his vehicles had 36 wheels. How many of each did he have?
8. How many integers are greater than or equal to 986 and strictly less than 1031?
9.
How many integers in the following sequence?
101, 111, 121, 131, .....3101
10.
In a single-elimination tournament, each team gets to play until it loses.
a) Suppose there are 16 teams in a single-elimination tournament. How many games will be
played in the tournament?
b) Suppose there are 21 teams. How many games will be played in the tournament?
c) Suppose there are 64 teams in the NCAA single-elimination tournament. How many
games will be played in the tournament?
d) Generalize for “n” teams. How many games will be played in the tournament?
11.
In a round-robin tournament, each team plays every other team exactly once.
a) Suppose there are 6 teams in a round-robin tournament. How many games will be played
in the tournament?
b) Suppose there are 11 teams in a round-robin tournament. How many games will be
played in the tournament?
c) Generalize for “n” teams. How many total games will be played in a round-robin
tournament?
12. Danny has a bunch of dice in his drawer. He knows there are 5 green dice, 6 blue dice and
7 red dice. He reaches in and grabs several without looking. How many dice must he grab in
order to ensure that he has grabbed 3 of the same color?
13. Find the shortest path from a to z in the weighted graph. State the sequence of vertices
in this shortest path and its length.
14.
A frog is at the bottom of a 10 meter well. Each day the frog jumps up 3 meters but at
night it slips down 2 meters. How many days will it take the frog to get out of the well?
15. Each person in a room shakes hands with every other person in the room. How many
handshakes occur if there are 8 people in the room?
16.
A ball rebounds ½ of the height from which it is dropped. Assume the ball is dropped
128 feet from the top of a building and keeps bouncing. How far will the ball have travelled
up and down when it strikes the ground for the 5th time?
17. I have a deck of cards from which some are missing. If I deal them equally to nine people,
I have two cards to spare. If I deal them equally to four people, I have three cards to spare.
If I deal them equally to seven people, I have five cards to spare. There are normally 52
cards in a full deck. Find the number of cards that are missing from this deck.
18. Jose needs to choose a new number for his locker combination. He can choose any
3-digit number using the digits from 1 to 5.
a) Assuming he can use a digit more than once, how many choices are there?
b) Assuming he can use a digit only once, how many choices does he have?
19. When the power returns after a power outage, an analog (traditional round) electric
Clock continues to measure time from the moment the power returns. However, a
digital clock returns to 12:00 and measures the amount of time that elapses after that.
When a man leaves his hours at 8:00 in the morning, his watch, round electric clock
and digital clock all agree. When he returns in the afternoon, his watch gives the time
as 3:20, his round electric clock reads 2:40 and his digital clock shows 1:30. At what
time did the power go off and when did the power come back on?