1. Where _precisely_ is the flaw in the following argument?
claim: Let x1, x2, ..., xn be any finite sequence of integers.
Then x1 = x2 = x3 = ... = xn (all the integers in the sequence are equal).
proof: By induction on n.
base case: Clearly the claim is true for n=1.
induction step. For n>1.
Assuming the claim is true for sequences of length n-1,
we show it is true for sequences of length n.
Let x1,x2,...,xn be any sequence of n integers.
Consider the two sequences
- x1,x2,...,xn-1
- and x2,x3,...,xn.
Each of these sequences has length n-1, so by induction
- x1 = x2 = ... = xn-1
- and x2 = x3 = ... = xn.
But since x2 is in both sequences, this implies
- x1 = x2 = ... = xn-1 = xn.
2. Consider an 8 by 8 checkerboard. Can you trace a path
from cell to cell on the board so that
- at each step the path moves from some cell to the cell directly above, below, to the left, or to the right and
- the path enters each white cell an even number of times, and enters each black cell an odd number of times
- the path finishes in the same cell it started?
Prove or disprove.
3. Each of n people know a distinct piece of gossip.
There are a sequence of phone calls. In each phone
call, one person tells all the gossip they know (up
to that point) to one other person. What's the minimum
number of phone calls needed before all people
know all the gossip? (Prove matching upper and
lower bounds).