"Lecture" for Week 3/4

Introductory remarks
This week is labeled "3/4" to keep the numbers in synch as we stretch out each week to a week-and-a-third during this period. Recall that Egypt/Babylon homework is due Wed. 9/17 and Early Greek homework is due Fri. 9/26.

Some apologies for matters that are beyond my control: First, I've posted a message from the system managers about the sluggish response of WebCT, which they hope will soon be remedied. Second, it has surely not escaped your notice that the online text is full of misprints. This is not surprising if Prof. Allen had to produce one chapter per week during the first semester he taught this course. I would like to add at least one chapter of my own about more modern aspects of mathematics, and I am wondering when I will find time to write even one chapter. (You see, I have this compulsion to proofread :-) Anyway, in the Greek chapter I noticed some misprints in the mathematics itself, or in the text in places that threaten to make it unintelligible. Fortunately, I have Heath's History of Greek Mathematics, so I should be able to clarify some points. (See below.)

Magic squares
Most of you successfully constructed a 3x3 magic square, and some made excellent observations, or developed methods, that taught me things about magic squares that I didn't know. Several people systematically listed all ways of writing 15 as the sum of 3 distinct integers and counted how many such triples each integer belongs to. From there it is easy to see that, if you adopt the "strong" definition of a magic square that requires the diagonals also to sum to 15, then the number in the middle square must be 5 and the numbers in the corners must be the even ones, and the remaining numbers fall into place. As one student noted, all strong magic squares of size 3 are basically the same; they are related by rotation and reflection (so there are 8 of them). Even in a weak magic square of size 3, the row and column containing integer p are always the same (although which of the two is the row may differ). So any two such squares are related by permutations of the rows, permutations of the columns, and interchanging rows with columns, which gives 72 cases by my count.

One student, and I, tried to prove the validity of the algorithm and almost succeeded. It is easy to show that the rows and columns add to the right number. The hard part is showing that each number appears in the square exactly once. We thought we had this, and that the generalization to composite n merely requires that p and q be relatively prime to n; but we discovered that the algorithm does not work right in the latter case. I hope to get back to you on this later, although it's not central to our syllabus.

Early Greek mathematics
Here are some things that were unclear to me until I read up on them in Heath's book.