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Induction
Ksenija Simic-Muller and Matt Ondrus Feb 28, 2007
Example 1. Every positive integer n can be expressed as n = c 0 + c 121 + c 222 + · · · + cM 2 M for some M ≥ 0, where ci ∈ { 0 , 1 } for all i.
Example 2. (Note: This is an example of a wrong proof.) Suppose that S : Z≥ 1 → Z is a function with the property that S(n) = 5S(n−1)− 6 S(n−2), where S(1) = 9 and S(2) = 20. Prove that S(n) = 3 · 2 n^ + 3n.
Example 3. Suppose you are given a square checkerboard with side-length 2n^ and with one missing square. Prove that the remaining squares on the board can be tiled with trominos.
Note that a tromino is an object shaped like.
Problem 1. A winning configuration in the game of MiniTetris is a complete tiling of a 2 × n board using only the three shapes shown below:
Prove that the number of winning configurations on a 2 × n MiniTetris board (n ≥ 1) is Tn = (2n+1^ + (−1)n)/3.
Problem 2. We are given a chocolate bar with m × n squares of chocolate, and our task is to divide it into mn individual squares. We are only allowed to split one piece of chocolate at a time using a vertical or a horizontal break. For example, suppose that the chocolate bar is 2 × 2. The first split makes two pieces, both 2 × 1. Each of these pieces requires one more split to form single squares. This gives a total of three splits. Use strong induction to conclude the following: Theorem. To divide up a chocolate bar with m × n squares, we need mn − 1 splits.
Problem 3. You begin with a stack of n boxes. Then you make a sequence of moves. In each move, you divide one stack of boxes into two nonempty stacks. The game ends when you have n stacks, each containing a single box. You earn points for each move; in particular, if you divide one stack of height a + b into two stacks with heights a and b, then you score ab points for that move. Your overall score is the sum of the points that you earn for each move. What strategy should you use to maximize your total score?
Problem 4. Prove that consecutive Fibonacci numbers are always relatively prime.
Problem 5. Show that every positive integer can be expressed uniquely as the sum of distinct, non-consecutive Fibonacci numbers (here, non-consecutive means that no two of the Fibonacci numbers in the sum are consecutive Fibonacci numbers).
Problem 6. Let n = 2k. Prove that we can select n integers from any (2n − 1) integers such that their sum is divisible by n.
Problem 7. Prove that if you triangulate a convex n-gon, then there are at least two vertices of degree two. Note: Think of a convex n-gon as a graph consisting of n vertices and n-edges arranged in a cycle. To triangulate an n-gon is to join non-adjacent vertices with edges in such a way that no edges cross each other and all of the resulting faces are triangles.
Problem 8. Let S(n) denote the number of strings of length n built from the alphabet {H, T } that do not contain the substring HH. Prove that
S(n) =
)n
)n .
Problem 9. Prove that the faces of a planar graph can be colored with two colors (so that no two adjacent faces are the same color) iff all of its vertices have even degree.
Problem 10. In an m × n matrix of real numbers, we mark at least p of the largest numbers (p ≤ m) in every column, and at least q of the largest numbers (q ≤ n) in every row. Prove that at least pq numbers are marked twice.
Problem 11. (Putnam, 1972) Show that, for all n > 1, n does not divide 2n^ − 1.
Problem 12. There are no positive integer solutions of x^4 + y^4 = z^2. Hint: You need to know that every positive integer solution of a^2 + b^2 = c^2 where a, b, and c are relatively prime can be expressed in terms of two relatively prime numbers m, and n where a = m^2 − n^2 , b = 2mn, and c = m^2 + n^2.
