The Pythagoreans noticed that nice simple ratios of string length
made nice sounds together.
Using an understanding that 1:2 and 2:3 were good ratios, start
with a length and keep reducing it to 2/3 of itself. Each time that
took the length under 1/2 they doubled it to get back within range.
The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.
Why is the modern piano tuned using an equal tempered scale and
what has this got to do with logarithms?
Use Euclid's algorithm to get a rational approximation to the
number of major thirds in an octave.
Show that it is rare for a ratio of ratios to be rational.
Ben and Ruth give proofs about the number of points with integer
coordinates on a parabola and a hyperbola.
Go to last month's problems to see more solutions.
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
This article sets some puzzles and describes how Euclid's algorithm
and continued fractions are related.
This sudoku requires you to have "double vision" - two Sudoku's for the price of one