Visualising and representing

A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.

Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

Make a cube out of straws and have a go at this practical challenge.

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

See if you can anticipate successive 'generations' of the two animals shown here.