Counting triangles on a tin ceiling (solution, take 2)

This is the solution to a problem I posed in my previous blog post. Please read that post first. [Update: I posted a solution to this problem two days ago. Yesterday Brent Yorgey pointed out that my analysis of “type 2” triangles was not quite correct. (See his comment below.) So I went back to…

Counting triangles on a tin ceiling

This morning a student popped in to my office and said he had a math question for me, and that it wasn’t related to a class he was taking. He said that his apartment has a tin ceiling with square tiles. The tiles have X’s through them, as illustrated below: He asked: “Is there a…

Sines and cosines (part 1)

A friend asked me the following question. What -values satisfy the equation ? It occurred to me to ask a more general question. Let  and . Let be the composition of with itself times. Similarly, let be the composition of with itself times. What are the solutions to ? Is it possible answer this question? Always? Sometimes? Never? I’ll post my…