A decorator can buy pink paint from two manufacturers. What is the
least number he would need of each type in order to produce
different shades of pink.
Is it always possible to combine two paints made up in the ratios
1:x and 1:y and turn them into paint made up in the ratio a:b ? Can
you find an efficent way of doing this?
The Egyptians expressed all fractions as the sum of different unit
fractions. The Greedy Algorithm might provide us with an efficient
way of doing this.
A jigsaw where pieces only go together if the fractions are
Can all unit fractions be written as the sum of two unit fractions?
Can you arrange these numbers into 7 subsets, each of three
numbers, so that when the numbers in each are added together, they
make seven consecutive numbers?
Five children went into the sweet shop after school. There were
choco bars, chews, mini eggs and lollypops, all costing under 50p.
Suggest a way in which Nathan could spend all his money.
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten
numbers from the bags above so that their total is 37.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second. . . .
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Think of two whole numbers under 10. Take one of them and add 1.
Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your
second number. Add 2. Double again. Subtract 8. Halve this. . . .
Can you see how to build a harmonic triangle? Can you work out the
next two rows?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
A circle of radius r touches two sides of a right angled triangle,
sides x and y, and has its centre on the hypotenuse. Can you prove
the formula linking x, y and r?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
What is the smallest number with exactly 14 divisors?
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. . . .
Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
A game for 2 or more people, based on the traditional card game
Rummy. Players aim to make two `tricks', where each trick has to
consist of a picture of a shape, a name that describes that shape,
and. . . .
The area of a square inscribed in a circle with a unit radius is,
satisfyingly, 2. What is the area of a regular hexagon inscribed in
a circle with a unit radius?
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G =
F and A-H represent the numbers from 0 to 7 Find the values of A,
B, C, D, E, F and H.
If you are given the mean, median and mode of five positive whole
numbers, can you find the numbers?
Explore the effect of reflecting in two parallel mirror lines.
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Each of the following shapes is made from arcs of a circle of
radius r. What is the perimeter of a shape with 3, 4, 5 and n
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Imagine you have a large supply of 3kg and 8kg weights. How many of
each weight would you need for the average (mean) of the weights to
be 6kg? What other averages could you have?
Can you find the area of a parallelogram defined by two vectors?
Can you maximise the area available to a grazing goat?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
If the hypotenuse (base) length is 100cm and if an extra line
splits the base into 36cm and 64cm parts, what were the side
lengths for the original right-angled triangle?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Explore the effect of combining enlargements.