Explaining, convincing and proving

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by 3, 5 and by 7...

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. Do you have any interesting findings to report?

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?

Which hexagons tessellate?

This grid can be filled so that each of the numbers 1, 2, 3, 4, 5 appears just once in each row, column and diagonal. Which number goes in the centre square?

Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?

If an athlete takes 10 minutes longer to walk, run and cycle three miles than he does to cycle all three miles, how long does it take him?