Creating and manipulating expressions and formulae

problem

Back to Basics

Age

14 to 16

Challenge level

Find b where 3723(base 10) = 123(base b).
problem

Square Pizza

Age

14 to 16

Challenge level

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
problem

Semi-Square

Age

14 to 16

Challenge level

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
problem

Poly Fibs

Age

16 to 18

Challenge level

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.
problem

Triangles Within Pentagons

Age

14 to 16

Challenge level

Show that all pentagonal numbers are one third of a triangular number.
problem

Integer Indices

Age

14 to 16

Challenge level

From this sum of powers, can you find the sum of the indices?
problem

Leftovers

Age

14 to 16

Challenge level

Weekly Problem 26 - 2008
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?
problem

Factor List

Age

14 to 16

Challenge level

Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?
problem

Why 8?

Age

11 to 14

Challenge level

Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own numbers. Why is the answer always 8?
problem

Multiply the Addition Square

Age

11 to 14

Challenge level

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?