Exploring and noticing

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

Is it the fastest swimmer, the fastest runner or the fastest cyclist who wins the Olympic Triathlon?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

How can we make sense of national and global statistics involving very large numbers?