### Approximating Pi

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

Which of these roads will satisfy a Munchkin builder?

### Twin Equivalent Sudoku

This Sudoku problem consists of a pair of linked standard Suduko puzzles each with some starting digits

# Spectrometry Detective

### Why do this problem?

This problem gives practice in mathematical reasoning in a genuine scientific context. It involves reasonably basic mathematics of combinatorics but requires more advanced inductive reasoning. It might prove an interesting context for the prospective mathematician and will definitely be useful for those interested in studying chemistry at university.

### Possible approach

As this problem does not fit directly into the typical chemistry or mathematics classroom it could be used as an end of term activity in which the focus is to encourage some clear mathematical thinking in a cross-curricular context.

Suggest that students interested in both chemistry and mathematics try the problem, either individually or in small groups. The basic chemical knowledge required is simply that of isotopes and atomic mass. The mass spectrometer might be unfamiliar, but should be simple to grasp in essence.

The focus throughout the problem should be on clarity in the mathematical explanations, and there will be an element of convincing others of the soundness of any resulting analysis.

When this problem was created it caused a great deal of discussions amongst students. Hopefully some of this discussion might be replicated amongst your students.

Finally, note that this problem is of an industrial, real world sort. It gives a flavour of the types of real questions which might be asked to professional scientists and mathematicians, where errors in mathematical reasoning can be highly costly . It gives great practice in such thinking.

### Key questions

Have you understood the chemical terms?

Which numbers seem to stand out, when compared with the periodic table?

Are you clear as to which aspects of you calculations are mathematically certain?

### Possible extension

Extension is built into this problem directly.

### Possible support

To make this problem more straightforward, provide copies of the periodic table. Suggest that students write down all the diatomic gasses that they know and start from these.

For a more straightforward foray into combinatoric chemistry, try the problem Heavy Hydrocarbons.