Math 497 Assignment #2

Textbook: The Book of Numbers by Conway and Guy (will be abbreviated BON)

Reading: Selected pages Chapter 3 of BON, especially pp. 65-74, 76-83.

Learning Assignment: 

You should learn and be able to reproduce, discuss and/or use a reasonable part of what is explored in class.

Problem Assignment:

1.1    Find and explain a couple of examples of counting with cards or dice or other situations and show how factorials or choice numbers enter into the answer.

1.2    A great circle divides a sphere into 2 hemispheres.  One special property of a great circle is that any two great circles intersect in two points (on opposite positions on the sphere).  Use a ball or and rubber bands to approximate a sphere and great circles.  For n = 1, 2, 3, 4, 5, 6 great circles, see how many pieces the sphere is cut into by n great circles. Make a table of your results. Then conjecture a formula if you can.  After this, use the difference table method to find a formula and check the formula on the next case or two.

1.3    Make a graphical Pascal triangle with at least 32 rows with the numbers in squares on graph paper.  But the arithmetic is easy.  Start with the 1's as usual, but use the rule of 0+0 = 1+1 = 0 and 0+1 = 1+0 = 1.  One way to think of this is that you are just keeping track of "odd" or "even" with 0 representing even and 1 representing odd.  Now color the squares with 1 dark and leave the squares with 0 light.  Can you explain the pattern?

1.4    Given numbers M_k, k = 0, 1, 2, 3, …

Then let S_N = M₀ + M₁ + M₂ + M₃ + M₄ + …  + M_N, for N = 0, 1, 2, …

Make a table of the S values and then find a polynomial P(N) so that P(N) = S_N for all N. (Use the finite difference method.)