Conjecturing and generalising

problem
Ball bearings

Age

16 to 18

Challenge level

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
problem
Absurdity again

Age

16 to 18

Challenge level

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
problem
Grouping goodies

Age

5 to 7

Challenge level

Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had?
problem
Three squares

Age

5 to 11

Challenge level

What is the greatest number of squares you can make by overlapping three squares?
problem
Lost books

Age

7 to 11

Challenge level

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?
problem
Let's investigate triangles

Age

5 to 7

Challenge level

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
problem
Making boxes

Age

7 to 11

Challenge level

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
problem
Sticky triangles

Age

7 to 11

Challenge level

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
problem
Magic constants

Age

7 to 11

Challenge level

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
problem
Round and round the circle

Age

7 to 11

Challenge level

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.