### Be Reasonable

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

### The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

### Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

# Spirostars

##### Stage: 5 Challenge Level:

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 $p$.
Variants of this activity are provided by using a circular geoboard, or in this interactivity which is a virtual geoboard, or by drawing on a printed sheet with dots around the circumferences of circles

All these activities presuppose that the points lie on a circle and the angle of turn is a rational multiple of 360 degrees and in this problem we make neither assumption.

This problem leads on to more general questions about spirolaterals. In the question Spiroflowers we consider what happens if the lengths of the line segments in the path are varied and what happens if both the lengths of the line segments and the angles of turn are varied.