Number theory

problem

Really Mr. Bond

Age

14 to 16

Challenge level

Can you explain what's happening with these numbers?
problem
Favourite

Always Perfect

Age

14 to 18

Challenge level

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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About Pythagorean Golden Means

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?
problem

Pythagorean Golden Means

Age

16 to 18

Challenge level

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.