Problem one was solved by 70% of the pupils. Problem 2 was solved
by 60% of them. Every pupil solved at least one of the problems.
Nine pupils solved both problems. How many pupils took the exam?
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are
natural numbers and 0 < a < b < c. Prove that there is
only one set of values which satisfy this equation.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Suppose you had to begin the never ending task of writing out the
natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the
1000th digit you would write down.
Ranging from kindergarten mathematics to the fringe of research
this informal article paints the big picture of number in a non
technical way suitable for primary teachers and older students.
What is the last digit of the number 1 / 5^903 ?
A cinema has 100 seats. Show how it is possible to sell exactly 100
tickets and take exactly £100 if the prices are £10 for
adults, 50p for pensioners and 10p for children.
Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Work out how to light up the single light. What's the rule?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
If you take a three by three square on a 1-10 addition square and
multiply the diagonally opposite numbers together, what is the
difference between these products. Why?
This package contains a collection of problems from the NRICH
website that could be suitable for students who have a good
understanding of Factors and Multiples and who feel ready to take
on some. . . .
This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
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?
There are some water lilies in a lake. The area that they cover doubles in size every day. After 17 days the whole lake is covered. How long did it take them to cover half the lake?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3
grid so that all the rows and columns add up to a prime number. How
many different solutions can you find?
What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
What is the smallest number with exactly 14 divisors?
This challenge is to make up YOUR OWN alphanumeric. Each letter
represents a digit and where the same letter appears more than once
it must represent the same digit each time.
Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!
Can you make a hypothesis to explain these ancient numbers?
Find out about palindromic numbers by reading this article.
How many positive integers less than or equal to 4000 can be
written down without using the digits 7, 8 or 9?
Whenever two chameleons of different colours meet they change
colour to the third colour. Describe the shortest sequence of
meetings in which all the chameleons change to green if you start
with 12. . . .
Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?
A combination mechanism for a safe comprises thirty-two tumblers
numbered from one to thirty-two in such a way that the numbers in
each wheel total 132... Could you open the safe?
Can you find any perfect numbers? Read this article to find out more...
This task depends on learners sharing reasoning, listening to
opinions, reflecting and pulling ideas together.
Which numbers can we write as a sum of square numbers?
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Can you work out how many of each kind of pencil this student
Take three whole numbers. The differences between them give you
three new numbers. Find the differences between the new numbers and
keep repeating this. What happens?
Robert noticed some interesting patterns when he highlighted square
numbers in a spreadsheet. Can you prove that the patterns will
Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two
digit numbers are multiplied to give a four digit number, so that
the expression is correct. How many different solutions can you
N people visit their friends staying N kilometres along the coast.
Some walk along the cliff path at N km an hour, the rest go by car.
How long is the road?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
When the number x 1 x x x is multiplied by 417 this gives the
answer 9 x x x 0 5 7. Find the missing digits, each of which is
represented by an "x" .