H OMEWORK 2
M ATH 13 FALL 2015
1. There is a unique solution to the KenKen puzzle shown in Figure 1. Pick one of the 16 squares
in this diagram, determine what its value is in the unique solution to the KenKen puzzle, and
carefully prove your answer is correct. (Do not solve the entire puzzle!)
F IGURE 1. This is a KenKen puzzle.
2. a. Let x, y 2 R. Using cases, prove that |x+y|  |x|+|y|. This is called the triangle inequality.
b. Find necessary and sufficient conditions on x, y for |x + y| = |x| + |y|.
c. Let a, b, c denote three points in R2 , and let d( , ) denote the distance between two points
in R2 . The two-dimensional version of the triangle inequality states that
d(a, c)  d(a, b) + d(b, c).
Draw a picture to help explain what this two-dimensional triangle inequality is asserting.
(Note that you are not asked to prove this two-dimensional triangle inequality.)
d. Find necessary and sufficient conditions on the points a, b, c for
d(a, c) = d(a, b) + d(b, c).
You don’t have to prove your answer.
e. What is the formula for d(a, b) in terms of the x and y-coordinates of a and b? Make sure to
clearly explain any notation you use.
f. Assume a, b, c 2 R2 all have y-coordinate equal to zero. Rewrite the two-dimensional triangle inequality in terms of the x-coordinates of a, b, c. What is the relationship between this
and the triangle inequality in part (a) above?
3. Consider a new variant on our board game: We have a sequence of squares extending infinitely
to the right, and a coin is in one of the squares. Two players take turns moving the coin left
either one or two spaces. A player wins (!!) if she/he moves the coin off the board.
1 2 3 4 5 6 7 8 9 10 11 12 · · ·
From which squares numbered 1-10 does player A have a winning strategy? (You don’t have to
prove your answer or show any work on this problem.)
4. Consider the sentence “The integers x and y are both greater than or equal to 10.” What is the
negation of this sentence? Write it in a way that doesn’t use the word “not”.
5. a. Prove or disprove: If an integer n is not divisible by 3, then there exists an integer k such that
n = 3 · k + 1.
b. Assume n is not divisible by 3. Prove that n2
one case is that n = 3 · k + 1.)
1 is divisible by 3. (Hint. Use cases, where
6. Using a truth table, show that
is logically equivalent to
P ) (Q ^ R)
(P ) Q) ^ (P ) R).
Write out what these two statements mean using words like “implies”, “or”, “and”, and “not”.
(You don’t have to use all of those words.) Does it seem reasonable that the two statements are
logically equivalent?
7. Is
a tautology? Explain.
(P ^ Q) ) (P _ Q)