If we take a number and square it, the answer is also the product of the two numbers either side of it plus one. Prove algebraically that this works for all numbers.

First of all let's give some examples: 5^2=4x6+1 = 25 or 6^2 = 5x7+1 = 36. To prove it for every number let's call the number x (e.g. x was 5 and 6 in the cases above). Then the result we want to prove is x^2= (x+1)(x-1)+1 (e.g. 5+1=6,5-1=4 in the first case). By expanding the right hand side (multiplying each term in each bracket) we get x^2 + x - x - 1 + 1, which simplifies to x^2. Since this is the same as the left hand side, we've proved the result algebraically.