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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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.
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?
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?
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. . . .
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?
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?
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. . . .
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you explain how this card trick works?
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?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
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.
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
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.
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
Delight your friends with this cunning trick! Can you explain how
A collection of games on the NIM theme
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. . . .
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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?
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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.
A game for 2 players
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Can you find the values at the vertices when you know the values on
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
It would be nice to have a strategy for disentangling any tangled
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?