Expanding and factorising quadratics

problem
Favourite

Iff

Age

14 to 18

Challenge level

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
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.
problem
Favourite

Parabella

Age

16 to 18

Challenge level

This is a beautiful result involving a parabola and parallels.
problem

Two Cubes

Age

14 to 16

Challenge level

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]
problem

Code to Zero

Age

16 to 18

Challenge level

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.
problem

Novemberish

Age

14 to 16

Challenge level

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.
problem

Fibonacci Factors

Age

16 to 18

Challenge level

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
problem

Multiplication Magic

Age

14 to 16

Challenge level

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.
problem

Never Prime

Age

14 to 16

Challenge level

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
problem

Composite Notions

Age

14 to 16

Challenge level

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.