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Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

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Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

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I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized one has radius 9 cm and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one?


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

Many students used trial and improvement techniques to show that to blend the coffee then 4kg of Mocha, 2 kg of Kenyan and 5 kg of Brazilian are needed if the conditions are to be satisfied. These included students from Maidstone Girls Grammar School; West Flegg Middle School; Madras College; Strabane Grammar School; Trinity Academy, Edinburgh; Our Lady's Grammar School, Newry; Clevedon Community School, Somerset and ACS Barker, Singapore.

Joshua of Russell Lower Scholl, Ampthill, Bedfordshire arrived at his solution which his father called trial and error techniques. There is an easier and quicker way submitted by Lyndsay of Mount school, York, which is represented below:

M + K + B = 11 (11 kg in all)
13M + 14K + 17B = 165 (cost is £165)
so 13 ( 11 - K - B) + 14K + 17B = 165
143 - 13K - 13B + 14K + 17B = 165
K + 4B = 22

Since B > K, the only whole numbers that fit are B = 5 and K = 2
So the answer is 5 kg Brazilian, 2 kg Kenyan and 4 Kg Mocha.