this month, Stage 3 & 4

April 2001

 

Problems

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Spirals and Wallpapers

Stage:3 Challenge Level:Challenge Level:1

A spiral is defined as: ...a curve formed by a point moving around a fixed point and constantly moving away from or approaching the fixed point. (p110. HBJ Dictionary of Mathematics compiled by Ken. . . .

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Is it Magic or Is it Maths?

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

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Making Rectangles, Making Squares

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

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3388

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

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Hot Pursuit

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

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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Nine Colours

Stage:3 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

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Square Mean

Stage:4 Challenge Level:Challenge Level:1

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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Qqq..cubed

Stage:4 Challenge Level:Challenge Level:1

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of. . . .

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AMGM

Stage:4 Challenge Level:Challenge Level:1

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

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Tricircle

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .

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Parabolic Patterns

Stage:4 and 5 Challenge Level:Challenge Level:1

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.