this month, Stage 3 & 4

June 2001

 

Problems

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2D-3D

Stage:3 Challenge Level:Challenge Level:1

Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of. . . .

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Bull's Eye

Stage:3 Challenge Level:Challenge Level:1

What fractions of the largest circle are the two shaded regions?

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Muggles Magic

Stage:3 Challenge Level:Challenge Level:1

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

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DOTS Division

Stage:4 Challenge Level:Challenge Level:1

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Our Ages

Stage:4 Challenge Level:Challenge Level:1

I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .

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Schlafli Tessellations

Stage:3, 4 and 5 Challenge Level:Challenge Level:1

are somewhat mundane they do pose a demanding challenge in terms of 'elegant' LOGO procedures. This problem considers the eight semi-regular tessellations which pose a demanding challenge in terms of. . . .

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Gnomon II

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

A gnomon is an L shape. Each Fibonacci number has its gnomon with area corresponding to the number. Draw dotted lines dividing each gnomon into two parts illustrating the Fibonacci rule.

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There's a Limit

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

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

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Ellipses

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

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

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Parabella

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

This is a beautiful result involving a parabola and parallels.