Conjecturing and generalising

problem

Patterns of Inflection

Age

16 to 18

Challenge level

Find the relationship between the locations of points of inflection, maxima and minima of functions.
problem

Speedy Summations

Age

16 to 18

Challenge level

Watch the video to see how to add together an arithmetic sequence of numbers efficiently.
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

Polite Numbers

Age

16 to 18

Challenge level

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?
problem

Fix Me or Crush Me

Age

16 to 18

Challenge level

Can you make matrices which will fix one lucky vector and crush another to zero?
problem

Irrational Arithmagons

Age

16 to 18

Challenge level

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
problem

Few and Far Between?

Age

16 to 18

Challenge level

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?
problem

Interpolating Polynomials

Age

16 to 18

Challenge level

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.