Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
How many different sets of numbers with at least four members can
you find in the numbers in this box?
What is the smallest number with exactly 14 divisors?
Ben’s class were making cutting up number tracks. First they
cut them into twos and added up the numbers on each piece. What
patterns could they see?
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
There are ten children in Becky's group. Can you find a set of
numbers for each of them? Are there any other sets?
Can you find a way to identify times tables after they have been shifted up?
I am thinking of three sets of numbers less than 101. They are the
red set, the green set and the blue set. Can you find all the
numbers in the sets from these clues?
Nearly all of us have made table patterns on hundred squares, that
is 10 by 10 grids. This problem looks at the patterns on
differently sized square grids.
I am thinking of three sets of numbers less than 101. Can you find
all the numbers in each set from these clues?
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some. . . .
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
Work out Tom's number from the answers he gives his friend. He will
only answer 'yes' or 'no'.
How many different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
Katie and Will have some balloons. Will's balloon burst at exactly
the same size as Katie's at the beginning of a puff. How many puffs
had Will done before his balloon burst?
Find the words hidden inside each of the circles by counting around
a certain number of spaces to find each letter in turn.
Norrie sees two lights flash at the same time, then one of them
flashes every 4th second, and the other flashes every 5th second.
How many times do they flash together during a whole minute?
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? Why?
The planet of Vuvv has seven moons. Can you work out how long it is
between each super-eclipse?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
The number 8888...88M9999...99 is divisible by 7 and it starts with
the digit 8 repeated 50 times and ends with the digit 9 repeated 50
times. What is the value of the digit M?
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?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
This article for teachers describes how number arrays can be a
useful reprentation for many number concepts.
A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.
How can you use just one weighing to find out which box contains
the lighter ten coins out of the ten boxes?
Find the highest power of 11 that will divide into 1000! exactly.
The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?
Can you find a relationship between the number of dots on the
circle and the number of steps that will ensure that all points are
In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
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
What is the lowest number which always leaves a remainder of 1 when
divided by each of the numbers from 2 to 10?
A game that tests your understanding of remainders.
These red, yellow and blue spinners were each spun 45 times in
total. Can you work out which numbers are on each spinner?
An investigation that gives you the opportunity to make and justify
I throw three dice and get 5, 3 and 2. Add the scores on the three
dice. What do you get? Now multiply the scores. What do you notice?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
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. . . .
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
Does a graph of the triangular numbers cross a graph of the six
times table? If so, where? Will a graph of the square numbers cross
the times table too?
A challenge that requires you to apply your knowledge of the
properties of numbers. Can you fill all the squares on the board?