Conjecturing and generalising

problem
2001 spatial oddity

Age

11 to 14

Challenge level

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
problem
Adding in rows

Age

11 to 14

Challenge level

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
problem
Where can we visit?

Age

11 to 14

Challenge level

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
problem
Why 24?

Age

14 to 16

Challenge level

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
problem
What's possible?

Age

14 to 16

Challenge level

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
problem
Oddly

Age

7 to 11

Challenge level

Find the sum of all three-digit numbers each of whose digits is odd.
problem
Cut it out

Age

7 to 11

Challenge level

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
problem
Quick times

Age

11 to 14

Challenge level

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.
problem
Fair shares?

Age

14 to 16

Challenge level

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
problem
Of all the areas

Age

14 to 16

Challenge level

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?