Assume you have a polynomial $p(x)$ of degree $n$ with roots $a_1, a_2, \dots, a_n$.
Let $S$ be the set of all roots. Let $S_i$ be the set of all subsets of $S$ with exactly $i$ elements. For example, if $S = \{ 1, 2, 3 \}$ then $S_2 = \{ \{1, 2\}, \{1, 3\}, \{2, 3\} \}$.
Let $\alpha_i = (-1)^i\Sigma\{ \Pi(y) \mid y \in S_i \}$.
Then
\[ p(x) = x^n + \alpha_1x^{n-1} + \alpha_2x^{n-2} + \cdots + \alpha_{n-1}x + \alpha_n \]
Just a small logic puzzle that I wrote:
Graduate Interviews
Six friends have just graduated from college and applied for a job at the same company. All six are applying for different jobs (one is in Purchasing) and hold different degrees (one is in Music). The three people interviewing them have scheduled them each for a private interview in either the morning or afternoon in one of three rooms. The nine people (six applicants, three interviewers) are five men (Abe, Charles, Ed, Hal, and Ivan) and four women (Becky, Darla, Fawn, and Gina). Determine who the interviewers were, which two people each of them interviewed, what job those people were applying for, and what degree each applicant holds.
1. If two men had the same interviewer, that person was also a man. No one was in the same room more than once.
2. The six people present at interviews in the morning were Gina, Fawn, the man with a Physics degree (who is not ivan), the one applying for a job in finance, and the two men using the conference room.
3. Becky came to an interview straight from lunch. Ed arrived after Ivan left.
4. The three job seekers in the afternoon were the one with a degree in mathematics, the one who interviewed for a job in Operations, and the one who interviewed in the Library.
5. Three of the four women are the one involved in the interview for the Public relations job, the one with an English degree, and the one who interviewed the psychology degree holder.
6. Fawn spent the afternoon in an interview with Charles (which was conducted in the corner office).
7. The interview of the art degree holder happened after the interview for the director of sales position. The woman seeking a job as a librarian was not interviewed in the library.
A quick way to estimate the number of sudoku solutions:
First, defining terminology:
A sudoku grid is a 9×9 grid broken up into 9 3×3 grids. Let A, B, C, D, E, F, G, H, J be 3×3 grids then
A B C
D E F
G H J
is a sudoku board. Letter each grid the same using lower case letters. Thus Aa is the top left corner of the sudoku board and Jj is the bottom right corner.
Now, some assumptions that reduce the number of solutions to be calculated:
As it turns out, this underestimates the real number by less than 4% (because the first assumption isn’t true). Here’s the calculation:
Number of ways to choose grid B given the above assumptions:
3 * 9 = 27 + 1 = 28
Now, the top three rows have been partitioned. Permitting permutations of the second and third row of grids B and C gives 6 * 6 * 6 * 6 = 1296 orderings for each of the 28 partitionings. This gives 28 * 1296 = 36,288 different possibilities for grids A, B, C. Taking advantage of the symmetry of the sudoku board, there are also 36,288 possibilities for A, B, G. Crossing the two, gives 36288 * 36288 = 1,316,818,944 possibilities for the corner grids. It was assumed that each of these would lead to four solutions:
1,316,818,944 * 4 = 5,267,275,776 solutions.
Ed Russel and Frazer Jarvis worked out the actual number as 5,472,730,538. You can see how here.
I would recommend the recently republished Formal Knot Theory
Ideally, authors should write with a target audience in mind. If a writer is skillful, this will increase the enjoyment of the reader. It is, perhaps, nowhere more important to keep the audience in mind than when writing mathematics.
The more mathematically sophisticated the intended reader, the more information which can be communicated in a short space. Consider the difference beween
\[ {{10}\choose{2}}\qquad \frac{10!}{2!8!}\qquad 45 \]
The expressions decrease in clarity of intent from left to right. The binomial (the left-hand expression) expresses clearly that the result is the number of pairs possible from a grouping of 10 choices. The central expression is slightly more ambiguous as to intent: it might be choosing 2 of 10, 8 of 10, or not the result of a choice at all. The right expression is devoid of intent.
However, the expressions increase in clarity of value from left to right. Less and less mathematical knowledge or calculation is required to discern the basic value of the expression.
It is a tension between these two goals, clarity of intent and clarity of value, that makes writing mathematics, aside from doing mathematics, such a difficult endeavor.