Homework 1, Mon 08-29-11
CS 2050, Intro Discrete Math for Computer Science
Due Wed 09-07-11 (Mon 09-05-11 is Labor Day)
Problem 1
Let n and m be integers.
(a) Prove that if at least one of n or m is even then the product n × m is even.
(b)Prove that if both n and m are odd then the product n × m is odd.
Problem 2 P
(a) Let Sn = ni=1 i. Prove that, for every n ≥ 2, Sn > n.
(b) Prove that there is unique positive integer that equals the sum of the positive integers not
exceeding it.
Problem 3
P
Let Sn be the sum of all positive integers from 1 to n, ie Sn = 1+2+. . .+n or Sn = ni=1 i.
Let Sn0 be the sum of the squares
all positive integers from 1 to n,
Pof
n
2
2
2
0
0
ie Sn = 1 +2 + . . . +n or Sn = i=1 i2 .
Let
Sn00 = 1×2 + 2×3 + 3×4 + . . . + n(n+1)
n
X
=
i(i+1) .
i=1
Prove that Sn00 = Sn + Sn0 .
Problem 4
P
i
Prove that, for every positive integer n, 2n
i=1 (1 + (−1) ) = 2n.
Problem 5
(a) Prove that the 5 × 5 board with the top left corner removed can be covered using 2 × 1 tiles.
(b) Prove that, for any odd integers n > 1 and m > 1, the n × m board with the top left corner
removed can be covered using 2 × 1 tiles.
Problem 6: Extra Credit
Let n > 1 and m > 1 be odd integers. Let S be the set of squares of the n × m board. Give a
complete characterization of the (single) squares that, if removed from the n × m board, then the
remaining (n×m)−1 area can be covered using 2×1 tiles. That is, characterize the set T ⊆ S such
that
x∈T
x∈S \T
=⇒
=⇒
S \ {x} can be covered using 2×1 tiles
S \ {x} cannot be covered using 2×1 tiles
1