Conjecturing and generalising

problem
Chord

Age

16 to 18

Challenge level

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.
problem
Make 37

Age

5 to 11

Challenge level

Four bags contain a large number of 1s, 3s, 5s and 7s. Can you pick any ten numbers from the bags so that their total is 37?
problem
Pick's theorem

Age

14 to 16

Challenge level

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
problem
Take three from five

Age

11 to 16

Challenge level

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
problem
Three times seven

Age

11 to 14

Challenge level

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
problem
Hidden rectangles

Age

11 to 14

Challenge level

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
problem
Threesomes

Age

11 to 14

Challenge level

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
problem
Which is quicker?

Age

7 to 11

Challenge level

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?
problem
Largest product

Age

11 to 14

Challenge level

Which set of numbers that add to 100 have the largest product?
problem
Got it

Age

7 to 14

Challenge level

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.