Creating and manipulating expressions and formulae

Think of any whole number. Each time you perform a sequence of operations on it, what do you notice about the divisors of your answer?

Can you show this algebraic expression is divisible by 4?

Using the new operator $\oplus$, can you solve this equation?

How does the perimeter change when we fold this isosceles triangle in half?

Look at the times that Harry, Christine and Betty take to pack boxes when working in pairs, to find how fast Christine can pack boxes by herself.

Can you explain how this card trick works?

Can you explain why a sequence of operations always gives you perfect squares?

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?