Volume 53


Dear Readers, welcome to Parabola!
This issue, published near year's end, reflects at the past and looks to the future.

This survey presents several constructions of a regular pentagon inscribed in a given circle together with proofs, old and new, of their correctness.

We show that Old Babylonian problem tablets contain a geometric proof of the irrationality of $\sqrt{2}$ predating the Greek discovery of this profound mathematical fact by more than a millennium.

The traditional Collatz Conjecture states that, for any number, if you divide by 2 if the number is even and, if odd, then multiply by 3 and add 1, and repeat, you will eventually reach 1.

An odd comic about even numbers.

Q1541 Consider $29x+30y+31z=366$ where $x,y,z$ are positive integers with $x<y<z$.
 (a) Without writing or using a computer, find such $x,y,z$.
 (b) Prove that there is only one solution.

Q1531 Take any four consecutive whole numbers, multiply them together and add 1.
Make a conjecture and prove it!

The problems from the 56th UNSW School Mathematics Competition.

The winners of the 56th UNSW School Mathematics Competition. Well done to you all!