## Problems

##### Age 5 to 7 Challenge Level:

Find all the numbers that can be made by adding the dots on two dice.

##### Age 5 to 7 Challenge Level:

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5,
6, 7...?

##### Age 7 to 11 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?

##### Age 7 to 11 Short Challenge Level:

56 406 is the product of two consecutive numbers. What are these
two numbers?

##### Age 7 to 11 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.

##### Age 11 to 14 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. . . .

##### Age 11 to 14 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. . . .

##### Age 11 to 14 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. . . .

##### Age 11 to 14 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?

##### Age 14 to 16 Challenge Level:

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

##### Age 14 to 16 Challenge Level:

What remainders do you get when square numbers are divided by 4?

##### Age 14 to 16 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. . . .

##### Age 14 to 16 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

##### Age 14 to 16 Challenge Level:

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

##### Age 16 to 18 Challenge Level:

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

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