Problems
Newspapers
Stage:2 Challenge Level:
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
Magazines
Stage:2 Challenge Level:
Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.
Home City
Stage:2 Challenge Level:
Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.
Football Sum
Stage:3 Challenge Level:
Find the values of the nine letters in the sum: FOOT + BALL = GAME
Permute It
Stage:3 Challenge Level:
Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in every possible order to give 5 digit numbers. Find the sum of the resulting numbers.
Quad in Quad
Stage:4 Challenge Level:
The points P, Q, R and S are the midpoints of the edges of a convex quadrilateral. What do you notice about the quadrilateral PQRS as the convex quadrilateral changes?
Prime Squares
Stage:4 Challenge Level:
Every prime number except 2, that is every odd prime number, is the difference of two squares. Prove that there is only one way to express an odd prime number as the difference of two squares. . . .
Fixing the Odds
Stage:4 Challenge Level:
You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two. . . .
Xtra
Stage:4 and 5 Challenge Level:
Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.
Belt
Stage:5 Challenge Level:
A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.
Powerful Factors
Stage:5 Challenge Level:
Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.
Tetra Perp
Stage:5 Challenge Level:
Show that the edges AD and BC of a tetrahedron ABCD are mutually perpendicular when: AB²+CD² = AC²+BD².
