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.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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?
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.
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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.
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.
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
An investigation that gives you the opportunity to make and justify
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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 two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
What happens when you round these three-digit numbers to the nearest 100?
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?
This activity involves rounding four-digit numbers to the nearest thousand.
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?
Find the sum of all three-digit numbers each of whose digits is
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?
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
What happens when you round these numbers to the nearest whole number?
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.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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 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
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?
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Can you explain how this card trick works?
This challenge asks you to imagine a snake coiling on itself.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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. . . .
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?