How many noughts are at the end of these giant numbers?

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

How many zeros are there at the end of the number which is the product of first hundred positive integers?

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

The harmonic triangle is built from fractions with unit numerators using a rule very similar to Pascal's triangle.

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

Compares the size of functions f(n) for large values of n.

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!