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Broad Topics > Algebra > Formulae

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For What?

Age 14 to 16 Challenge Level:

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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Sums of Squares

Age 16 to 18 Challenge Level:

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?

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On the Importance of Pedantry

Age 16 to 18

A introduction to how patterns can be deceiving, and what is and is not a proof.

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What's That Graph?

Age 14 to 16 Challenge Level:

Can you work out which processes are represented by the graphs?

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Training Schedule

Age 14 to 16 Challenge Level:

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?

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Quadratic Transformations

Age 14 to 16 Challenge Level:

Explore the two quadratic functions and find out how their graphs are related.

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More Quadratic Transformations

Age 14 to 16 Challenge Level:

Here are some more quadratic functions to explore. How are their graphs related?

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Whose Line Graph Is it Anyway?

Age 16 to 18 Challenge Level:

Which line graph, equations and physical processes go together?

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Pick's Quadratics

Age 16 to 18 Challenge Level:

Find a quadratic formula which generalises Pick's Theorem.

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First Forward Into Logo 12: Puzzling Sums

Age 11 to 18 Challenge Level:

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

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Screen Shot

Age 14 to 16 Challenge Level:

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. . . .

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Spread

Age 16 to 18 Challenge Level:

Given the mean and standard deviation of a set of marks, what is the greatest number of candidates who could have scored 100%?

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Triangles Within Pentagons

Age 14 to 16 Challenge Level:

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

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Triangles Within Squares

Age 14 to 16 Challenge Level:

Can you find a rule which relates triangular numbers to square numbers?

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Triangles Within Triangles

Age 14 to 16 Challenge Level:

Can you find a rule which connects consecutive triangular numbers?

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Double Time

Age 16 to 18 Challenge Level:

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

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Ordered Sums

Age 14 to 16 Challenge Level:

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. . . .

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Hot Pursuit

Age 11 to 14 Challenge Level:

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

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Icosagram

Age 11 to 14 Challenge Level:

Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at. . . .

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In Particular

Age 14 to 16 Challenge Level:

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.

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Pinned Squares

Age 11 to 14 Challenge Level:

The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .