Explaining, convincing and proving
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problemHiking the Hill
Sarah's average speed for a journey was 2 mph, and her return average speed was 4 mph. What is her average speed for the whole journey?
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problemThe London Eye
The 80 spokes of The London Eye are made from 4 miles of cable. What is the approximate circumference of the wheel?
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problemPeter's Primes
Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?
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problemDifferent Products
Take four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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problemProximity
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
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problemGift of Gems
Four jewellers share their stock. Can you work out the relative values of their gems?
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problemLong Short
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
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problemConverse
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?