What is the smallest number with exactly 14 divisors?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
Here's a chance to work with large numbers...
A spider is sitting in the middle of one of the smallest walls in a
room and a fly is resting beside the window. What is the shortest
distance the spider would have to crawl to catch the fly?
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. . . .
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
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. . . .
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?
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.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?
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 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
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.
Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
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
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?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Some people offer advice on how to win at games of chance, or how
to influence probability in your favour. Can you decide whether
advice is good or not?
Chris and Jo put two red and four blue ribbons in a box. They each
pick a ribbon from the box without looking. Jo wins if the two
ribbons are the same colour. Is the game fair?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
How many different symmetrical shapes can you make by shading triangles or squares?
Can you describe this route to infinity? Where will the arrows take you next?
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. . . .
Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .
Can you work out the dimensions of the three cubes?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Your school has been left a million pounds in the will of an ex-
pupil. What model of investment and spending would you use in order
to ensure the best return on the money?
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
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.