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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
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.
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.
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.
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you explain how this card trick works?
Delight your friends with this cunning trick! Can you explain how
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
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?
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?
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. . . .
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
It starts quite simple but great opportunities for number discoveries and patterns!
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.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
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. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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 an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Can you find sets of sloping lines that enclose a square?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
A game for 2 players
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
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
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?