Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Multiplication Square

Age 14 to 16 Challenge Level:

Take a look at the multiplication square below:

Pick any 2 by 2 square and add the numbers on each diagonal.
For example, if you take:

the numbers along one diagonal add up to $77$ ($32 + 45$)
and the numbers along the other diagonal add up to $76$ ($36 + 40$).

Try a few more examples.
What do you notice?
Can you show (prove) that this will always be true?

Now pick any 3 by 3 square and add the numbers on each diagonal.
For example, if you take:

the numbers along one diagonal add up to $275$ ($72 + 91 + 112$)
and the numbers along the other diagonal add up to $271$ ($84 + 91 + 96$).

Try a few more examples.
What do you notice this time?
Can you show (prove) that this will always be true?

Now pick any 4 by 4 square and add the numbers on each diagonal.
For example, if you take:

the numbers along one diagonal add up to $176$ ($24 + 36 + 50 + 66$)
and the numbers along the other diagonal add up to $166$ ($33 + 40 + 45 + 48$).

Try a few more examples.
What do you notice now?
Can you show (prove) that this will always be true?

Can you predict what will happen if you pick a 5 by 5 square, a 6 by 6 square ... an n by n square, and add the numbers on each diagonal?