Discrete Mathematics
Problem Sheet 1
Pigeon-hole Principle:
1. Use the pigeon-hole principle to show that one of any nconsecutive inte-
gers is divisible by n.
2. Use the pigeon-hole principle to show that the decimal expansion of a
rational number must, after some point, become periodic.
3. The circumference of two concentric disks is divided into 200 sections
each. For the outer disk, 100 of the sections are painted red and 100 of
the sections are painted white. For the inner disk the sections are painted
red or white in an arbitrary manner. Show that it is possible to align the
two disks so that 100 or more of the sections on the inner disk have their
colors matched with the corresponding sections on the outer disk.
4. Given 20 French, 30 Spanish, 25 German, 20 Italian, 50 Russian and 17
English books, how many books must be chosen to guarantee that at least
(a) 10 books of one language were chosen?
(b) 6 French, 11 Spanish, 7 German, 4 Italian, 20 Russian, or 8 English
were chosen?
5. If there are 104 different pairs of people who know each other at a party
of 30 people, then show that some person has 6 or fewer acquaintances.
6. Show that given any 52 integers, there exist two of them whose sum, or
else whose difference, is divisible by 100.
7. From the integers 1,2, . . . , 200, 101 integers are chosen. Show that among
the integers chosen there are two such that one of them is divisible by the
other.
8. A student has 37 days to prepare for an examination. From past experi-
ence she knows that she will require no more than 60 hours of study. She
also wishes to study at least 1 hour per day. Show that no matter how
she schedules her study time (a whole number of hours per day), there is
a succession of days during which she will have studied exactly 13 hours.
9. Prove that in a group of npeople there are two who have the same number
of acquaintances in the group.
10. Given the information that no human being has more than 300,000 hairs
on his head, and that the state of Florida has a population of 10,000,000,
observe that there are at least two persons in Florida with the same num-
ber of hairs on their head. What is the largest integer that can be used
for nin the following assertion? There are npersons in Florida with the
same number of hairs on their heads.
1
