Explaining, convincing and proving

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Show that all pentagonal numbers are one third of a triangular number.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Drawing a triangle is not always as easy as you might think!

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

When the numbers from 1 to 1000 are written on a blackboard, which digit appears the most number of times?

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

Which of these roads will satisfy a Munchkin builder?

Can you prove these triangle theorems both ways?