Noticing Patterns
The key to solving these problems is to notice patterns or properties. Encouraging students to organise their work systematically allows them to notice what might not otherwise be obvious.
Finding All Solutions
These problems challenge students to find all possible solutions. One of the best answers to "How do you know you have found them all" is to be able to say "I worked systematically!"
Consecutive Numbers
Age 7 to 14
Challenge Level
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Add to 200
Age 11 to 14
Challenge Level
By selecting digits for an addition grid, what targets can you make?
Multiples Sudoku
Age 11 to 14
Challenge Level
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
Sticky Numbers
Age 11 to 14
Challenge Level
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Cinema Problem
Age 11 to 14
Challenge Level
A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.
Gabriel's Problem
Age 11 to 14
Challenge Level
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Fence It
Age 11 to 14
Challenge Level
This problem challenges students to work systematically while applying their knowledge of areas of rectangles.
Shifting Times Tables
Age 11 to 14
Challenge Level
This problem encourages students to think about the properties of numbers. It could be used as an introduction to work on linear sequences and straight line graphs.
Number Daisy
Age 11 to 14
Challenge Level
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
Isosceles Triangles
Age 11 to 14
Challenge Level
This problem encourages students to use coordinates, area and isosceles triangles to solve a non-standard problem. To find all possible solutions they will need to work systematically.
Shady Symmetry
Age 11 to 14
Challenge Level
This problem is a good activity for the visualisation of symmetry, and for encouraging students to work systematically.
American Billions
Age 11 to 14
Challenge Level
This is an engaging context in which to reinforce rules of divisibility and challenge students to reason mathematically and work systematically.
Summing Consecutive Numbers
Age 11 to 14
Challenge Level
This problem offers a simple context for students to explore, make generalisations and prove conjectures, working numerically and algebraically.
Sociable Cards
Age 11 to 14
Challenge Level
Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?
Special Numbers
Age 11 to 14
Challenge Level
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
M, M and M
Age 11 to 14
Challenge Level
This problem offers the students an opportunity to consolidate what they are expected to know about mean, mode and median whilst also challenging them to work systematically, and justify their reasoning
Can They Be Equal?
Age 11 to 14
Challenge Level
Sometimes area and perimeter of rectangles are taught separately, and are often confused. In this problem students consider the relationship between them.
Where Can We Visit?
Age 11 to 14
Challenge Level
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?
Weights
Age 11 to 14
Challenge Level
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Squares in Rectangles
Age 11 to 14
Challenge Level
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?
1 Step 2 Step
Age 11 to 14
Challenge Level
This problem is inaccessible without looking at simpler cases, and thus helps students to see the value of specialising in order to generalise.
Peaches Today, Peaches Tomorrow...
Age 11 to 14
Challenge Level
This problem could replace repetitive textbook work on calculating fractions of integers. It offers plenty of practice of these calculations while requiring students to come up with problem-solving strategies.
Triangles to Tetrahedra
Age 11 to 14
Challenge Level
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Cuboids
Age 11 to 14
Challenge Level
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Consecutive Negative Numbers
Age 11 to 14
Challenge Level
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Working Systematically - Short Problems
Age 11 to 16
A collection of short Stage 3 and 4 problems on Working Systematically.
Charlie's Delightful Machine
Age 11 to 16
Challenge Level
This problem could be used to reinforce work on recording and describing linear sequences.
Product Sudoku
Age 11 to 16
Challenge Level
This problem offers an engaging context in which to apply knowledge of factors, multiples and prime factor decomposition.
Ben's Game
Age 11 to 16
Challenge Level
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
Nine Colours
Age 11 to 16
Challenge Level
Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?
Two and Two
Age 11 to 16
Challenge Level
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Shopping Basket
Age 11 to 16
Challenge Level
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
Warmsnug Double Glazing
Age 14 to 16
Challenge Level
This problem offers students a chance to develop strategies for organising and understanding mixed up information within the context of calculating areas and perimeters of rectangles.
Pick's Theorem
Age 14 to 16
Challenge Level
This problem allows students to consolidate their understanding of how to calculate the area of irregular shapes, while offering an opportunity to explore and discover an interesting result.
LCM Sudoku
Age 14 to 16
Challenge Level
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
What's Possible?
Age 14 to 16
Challenge Level
As well as introducing the difference of two squares, this problem allows students to explore, conjecture and use algebra to justify their results.
Difference Sudoku
Age 14 to 16
Challenge Level
Use the differences to find the solution to this Sudoku.
NRICH is part of the family of activities in the Millennium Mathematics Project.