Creating and manipulating expressions and formulae

problem
Folding in half

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

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.
problem
Card trick 1

Age

11 to 14

Challenge level

Can you explain how this card trick works?
problem
Regular hexagon loops

Age

11 to 14

Challenge level

Make some loops out of regular hexagons. What rules can you discover?
problem
Diophantine n-tuples

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
problem
There and back

Age

14 to 16

Challenge level

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?
problem
Series sums

Age

14 to 16

Challenge level

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

Age

11 to 14

Challenge level

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?
problem
Hallway borders

Age

11 to 14

Challenge level

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?