Find the sum of all three-digit numbers each of whose digits is
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
It would be nice to have a strategy for disentangling any tangled
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
This challenge asks you to imagine a snake coiling on itself.
Can you tangle yourself up and reach any fraction?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Can all unit fractions be written as the sum of two unit fractions?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
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?
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
Find out what a "fault-free" rectangle is and try to make some of
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
An investigation that gives you the opportunity to make and justify
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
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.
Can you dissect an equilateral triangle into 6 smaller ones? What
number of smaller equilateral triangles is it NOT possible to
dissect a larger equilateral triangle into?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
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?
Can you see how to build a harmonic triangle? Can you work out the
next two rows?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
A collection of games on the NIM theme
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.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
It starts quite simple but great opportunities for number discoveries and patterns!
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. . . .
Can you find sets of sloping lines that enclose a square?
How many different journeys could you make if you were going to
visit four stations in this network? How about if there were five
stations? Can you predict the number of journeys for seven
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Can you work out how to win this game of Nim? Does it matter if you
go first or second?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
One block is needed to make an up-and-down staircase, with one step
up and one step down. How many blocks would be needed to build an
up-and-down staircase with 5 steps up and 5 steps down?
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?