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A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Estimating Time

Why do this problem?

Estimating time is not a new idea for a data gathering exercise, but the tool provided in this problem allows easy collection of data that can be pasted into a spreadsheet, offering an ideal opportunity for some work on using spreadsheets to calculate statistics. See also the problem Reaction Timer.

Possible approach

You may wish to use the data in the problem Half a Minute to introduce the problem.

There are a wide range of questions that can be explored with the tool, and analysis can be performed at lots of different levels, so this is an ideal problem for all ages.

For younger students, you could draw out the idea of averaging repeated trials, and discuss which average might be most appropriate.

Older students might look at moving averages to see whether someone becomes better at estimating.

In order to compare two people's results, it might be necessary to introduce measures of spread - range and interquartile range, and for the oldest students, variance and standard deviation.

If different classes are working on this problem at the same time, there's a great opportunity for sharing data between them, and being able to work on bigger hypotheses like comparing students of different ages. If your school creates a large data set using this tool, please get in touch if you'd like to share it!

You could discuss different strategies that could be used to improve one's performance at estimating particular time intervals (such as reciting something at a steady speed or counting using a phrase exactly one second long). Then the class could design a statistical test to evaluate the different methods.