The numbers behind the music

I've spent this morning tutoring at a Year 8 Mathematics Masterclass on "Mathematics and Music", for which I have to admit to only having competency in one half thereof, but in spite of that it was very interesting. Towards the end of the lecture, my attention was drawn to the existence of the Fibonacci Waltz, a composition based on the Fibonacci sequence of numbers. How it is composed is by assigning a note to each number, so 1 is C, 2 is D and so on. Sharps are not considered for the purpose of this (although if someone wishes to attempt further composition...), and similarly octaves are disregarded so number 8 corresponds back to C, essentially meaning running the Fibonacci sequence in a form of modular 7 arithmetic. 

There are a couple of surprises in this piece, the first of which is that it is somewhat aesthetically pleasing! Mathematically though, I was surprised to see that it has a repeating sequence of 16 notes, and I'm now rather keen to see why. For a start, I would have probably expected the repeat to be some multiple of 7. Further to that, would the sequence repeat if sharps were included (turning it into a form of mod 11 arithmetic)? If it does, what about other modular bases? This is something that requires further investigation, I reckon, and I can already see an A-level class investigation brewing over this one...

The Great Wikipedia Race!

Introduced by the top set Year 11s and now the new craze in the staff room. A game for 2 or more players:

Each player logs on to a computer and goes to the Wikipedia homepage. Choose a destination article (such as Isaac Newton, Sydney Harbour Bridge, etc), then each player hits the "Random Article" button. The aim is then to navigate to the destination article as quickly as possibly by using only links found in the articles you are viewing, starting with the random article. 

Hours of amusement, and good use of the random tidbits and connections connected as a quizzer!

Oh yes, and I have indeed lost the game.

The Path of Least Expectation

It's been a while since I've posted. This is due to finding out first hand that January is a rather long month in the teaching profession (typified by the question asked by one of the science teachers when he asked if I could come up with a mathematical explanation of why August never drags by January always does...), but for the sake of providing at least one post a month, I'll offer this poser...

I get quite annoyed when classes ask to change their seating plan, so my response to a question of "when are we changing the seating plan?" is "when you least expect it", pointing out that because they asked, they must have been, to some extent, expecting me to cave and allow it to happen that day. Now this is just to annoy them (hey, I'm allowed...), but here's the thought:

If I'd said to them instead, "Sometime this term, but when you don't expect it.", would there be a time available in the term to change the seating plan? To demonstrate this, imagine that we got to the last lesson of term and I still hadn't changed the seating plan. In this case, to keep my word, I would have to change the seating plan. But the pupils, if they remembered my original statement, would be expecting it, because it would be my last chance so I wouldn't have fulfilled the condition in my statement. Now, think if I got to the second last lesson. In this instance, because I know the pupils would have to expect my change if I left it to the last lesson, and so to avoid my pupils' expectation I would have to change it this lesson. Of course, I teach some bright cookies, and they too would figure this out in the second last lesson.

This logical process can be applied to every lesson of the term by continuing the iteration to the third from last, fourth from last and so on, so my pupils have to resign themselves to the fact that I simply can't fulfil my statement and the seating plan seems doomed to stay. Or does it? Is there a flaw in the logic, and why/why not?