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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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 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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Are these statements always true, sometimes true or never true?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
This challenge asks you to imagine a snake coiling on itself.
Can you explain the strategy for winning this game with any target?
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
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.
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?
Here are two kinds of spirals for you to explore. What do you notice?
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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. . . .
This activity involves rounding four-digit numbers to the nearest thousand.
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
Got It game for an adult and child. How can you play so that you know you will always win?
Find the sum of all three-digit numbers each of whose digits is
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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. . . .
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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?
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?
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
An investigation that gives you the opportunity to make and justify
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?
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Can you find all the ways to get 15 at the top of this triangle of numbers?
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?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
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 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. . . .
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?