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Be Reasonable

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

Good Approximations

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

Rational Roots

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Spirostars

Age 16 to 18 Challenge Level:

20 point star
To produce this star, twenty line segments of equal length are drawn in a continuous path, with equal angles between consecutive line segments.

Imagine instructing a small creature to walk along the path. You would give the instruction to walk forward a certain distance then to turn through a certain angle and to repeat the instruction over and over again.

To do this, you could use the Logo commands:
repeat 20 [forward 100 right $\theta$]

Experiment with the Logo program
repeat $q$ [forward 100 right $\theta$]

What shapes can you draw? Vary $q$ and $\theta$. For what values of $\theta$ can you find closed paths (returning to the starting point)?

Prove that the path is closed if and only if $\theta$ is a rational multiple of 360 degrees.

Compare this property to the results found in the problem Stars.