Spirostars

This challenge extends other NRICH challenges which present the same mathematical system for exploration in different ways by different groups of learners: 'Round and round the circle', Path to the stars' and Stars. In the first and last of these you will find interactivities for experimenting with these ideas. In Stars there are downloadable pdfs so you can print out circles with different numbers of equally spaced dots around the circles for paper and pencil explorations.

Celia Hoyles once said that the best mathematical problems challenge people of all ages by providing some questions that everyone can answer, and aesthetically pleasing patterns that everyone can appreciate, while at the same time leading to more challenging questions and more general conjectures which require sophisticated arguments and rigorous proofs.

Lynne McClure introduced us to the activity where, given enough space and enough string, a whole class can stand in a circle and the string can be passed from one person to another to form a star. Suppose there are $q$ people in the circle and the string is passed to the $p$th person around the circle each time. This leads to many questions about symmetry, about how the stars are formed, about why the activity sometimes produces regular polygons, and about how many different stars can be formed for a given $q$ by varying $p$.