We assume the reader is familiar with the concepts of determinism and finite automata, or has read the corresponding primer on this blog. The Mother of All Computers Last time we saw some models for computation, and saw in turn how limited they were.

The first step in studying the sorts of possible computations (and more interestingly, those things which cannot be computed) is to define exactly what we mean by a “computation.” At a high level, this is easy: a computation is simply a function. Given some input, produce the appropriate output. Unfortunately this is much too general. For instance, we could define almost anything we want in terms of functions.

Problem: Prove that for all $ n,k \in \mathbb{N}, k > 1$, we have $$\sum \limits_{i=0}^{n} k^i = \frac{k^{n+1}-1}{k-1}$$ Solution: Representing the numbers in base $ k$, we have that each term of the sum is all 0’s except for a 1 in the $ i$th place. Hence, the sum of all terms is the $ n$-digit number comprised of all 1’s. Multiplying by $ k-1$ gives us the $ n$-digit number where every digit is $ k-1$.

Additional Patterns Last time we left the reader with the assertion that Conway’s game of life does not always stabilize. Specifically, there exist patterns which result in unbounded cell population growth. Although John Conway’s original conjecture was that all patterns eventually stabilize (and offered $50 to anyone who could provide a proof or counterexample), he was proven wrong.

Cellular Automata There is a long history of mathematical models for computation. One very important one is the Turing Machine, which is the foundation of our implementations of actual computers today. On the other end of the spectrum, one of the simpler models of computation (often simply called a system) is a cellular automaton. Surprisingly enough, there are deep connections between the two.

Problem: Take a chessboard and cut off two opposite corners. Is it possible to completely tile the remaining board with 2-by-1 dominoes? Solution: Notice that every domino covers exactly one white tile and one black tile. Counting up the colors, we have 32 white and 30 black. Hence, any tiling by 2-by-1 dominoes will leave two extra white squares unaccounted for. So no such tiling is possible. Problem: Cut one corner off a chessboard.

Community Service Mathematics is supposed to be a process of discovery. Definitions, propositions, and methods of proof don’t come from nowhere, although after the fact (when presented in a textbook) they often seem to. As opposed to a textbook, real maths is highly non-linear.

Problem: What is the area of the triangle within the rectangle? Solution: In a moment of inspiration, we draw the following additional line: Now the answer is obvious. Once we split the rectangle into two smaller rectangles, the sides of the triangle become diagonals of their respective rectangles. The diagonals obviously split each of the two smaller rectangles into halves, where one half lies inside our original triangle.

Problem: Find the sum of the first 1000 natural numbers. Solution: Write the numbers twice as follows: $ \begin{matrix} 1 & + & 2 & + & \dots & + & 999 & + & 1000 \\\ 1000 & + & 999 & + & \dots & + & 2 & + &

Problem: At any party of 1000 people, must there always exist two people at the party who have the same number of friends at the party? For the sake of this problem, one cannot be friends with oneself, and friendship is bidirectional. Solution: This must always happen. Suppose to the contrary, that every person at the party has a different number of friends at the party. The minimum number of friends one could have is 0, while 999 is the maximum.

Problem: 1000 players compete in a tournament. In each round, players are matched with opponents, and the winner proceeds to the next round. If there are an odd number of players in a round, one player chosen at random sits out of that round. What is the total number of games are played in the tournament? Solution: 999. Each player loses exactly one game, except for the winner of the tournament.