Explaining, convincing and proving

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage stamps? Prove that all other values can be made up.

A car is travelling along a dual carriageway at constant speed. Every 3 minutes a bus passes going in the opposite direction, while every 6 minutes a bus passes the car travelling in the same direction. Buses leave the depot at regular intervals; they travel along the dual carriageway and back to the depot at a constant speed. At what interval do the buses leave the depot?

Can you find a rule which connects consecutive triangular numbers?

How many bees could fly 1000 miles if they had 10 gallons of honey?

Sarah's average speed for a journey was 2 mph, and her return average speed was 4 mph. What is her average speed for the whole journey?

The 80 spokes of The London Eye are made from 4 miles of cable. What is the approximate circumference of the wheel?

Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?

Take four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?