Problems
Two Dice
Stage:1 Challenge Level:
Find ALL the numbers that can be made by adding the dots on two dice.
Making Rectangles
Stage:1 Challenge Level:
Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?
Doplication
Stage:2 Challenge Level:
We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?
Special 24
Stage:2 Challenge Level:
Find another number that is one short of a square number and when you double it and add 1, the result is also a square number.
What 2
Stage:2 Challenge Level:
56 406 is the product of two consecutive numbers. What are these two numbers?
Symmetricality
Stage:3 Challenge Level:
Add up all 5 equations given below. What do you notice? Solve the system and find the values of a, b, c , d and e. b + c + d + e = 4 a + c + d + e = 5 a + b + d + e = 1 a + b + c + e = 2 a + b. . . .
Why 8?
Stage:3 Challenge Level:
Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own. . . .
Flagging
Stage:3 Challenge Level:
How many tricolour flags are possible with 5 available colours such that two adjacent stripes must NOT be the same colour. What about 256 colours?
Tied Up
Stage:3 Challenge Level:
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
The Pillar of Chios
Stage:3 Challenge Level:
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Rotating Triangle
Stage:3 and 4 Challenge Level:
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
Coke Machine
Stage:4 Challenge Level:
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly. . . .
Overturning Fracsum
Stage:4 Challenge Level:
Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7
One and Three
Stage:4 Challenge Level:
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
Grid Lockout
Stage:4 Challenge Level:
What remainders do you get when square numbers are divided by 4?
Telescoping Series
Stage:5 Challenge Level:
Find S_r = 1^r + 2^r + 3^r + ... + n^r where r is any fixed positive integer in terms of S_1, S_2, ... S_{r-1}.
