Show that all pentagonal numbers are one third of a triangular number.
A introduction to how patterns can be deceiving, and what is and is not a proof.
Can you find a rule which connects consecutive triangular numbers?
Can you find a rule which relates triangular numbers to square numbers?
Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .
Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Given the mean and standard deviation of a set of marks, what is the greatest number of candidates who could have scored 100%?
Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.
Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.
Here are some more quadratic functions to explore. How are their graphs related?
The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?
Explore the two quadratic functions and find out how their graphs are related.
A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .
Find a quadratic formula which generalises Pick's Theorem.
Write 100 as the sum of two positive integers, one divisible by 7 and the other divisible by 11. Then find formulas giving all the solutions to 7x + 11y = 100 where x and y are integers.
Which line graph, equations and physical processes go together?
Can you work out which processes are represented by the graphs?