Math 2200 Homework 6
Due Date: October 30
Please write neatly and leave enough space so I can write comments! Remember that there are solutions
to all the odd-numbered exercises in the textbook. I will mainly be looking at your solutions to the evennumbered exercises from the textbook and all the exercises I write, so make sure you attempt most of those.
For formal proofs, please state the theorem being proved, write “proof”, and then give your argument using
complete English sentences. Please show your work on all problems!
Warm-up
Exercise 1. Use induction to prove the following theorem:
Theorem. Let n ∈ Z≥0 . Then the sum of the first n even positive integers is n(n + 1).
Exercise 2. A long time ago, we proved that the square of any odd integer is odd. Prove this for odd
positive integers by induction:
Theorem. Let n ∈ Z≥0 . Then (2n + 1)2 is odd.
Also give a one-line proof of this theorem using congruence modulo 2.
Section 5.1 (p. 329-332)
# 3, # 31, # 32, # 35, # 57. (Hint: use congruences for #31, #32, and # 35.)
Optional Difficult Riddle
Josephine’s Problem: In Josephine’s Kingdom every woman has to pass a logic exam before being allowed
to marry. Every married woman knows about the fidelity of every man in the Kingdom except for her own
husband, and etiquette demands that no woman should tell another about the fidelity of her husband. Also,
a gunshot fired in any house in the Kingdom will be heard in any other house. Queen Josephine announced
that unfaithful men had been discovered in the Kingdom, and that any woman knowing her husband to be
unfaithful was required to shoot him at midnight following the day after she discovered his infidelity. How
did the wives manage this?
(See Wikipedia’s article “Induction puzzles” for the solution and for more riddles.)
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