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Mathematicians often talk about the importance of working systematically. This means that rather than working in a haphazard and random way, there is a methodical, organised and logical approach. The problems below will challenge students to work systematically, and will help them appreciate the benefits of working in this way.
Scroll down to see our complete collection of problems, or explore the two sub-collections.
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.
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!"
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?
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?
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
Sometimes area and perimeter of rectangles are taught separately, and are often confused. In this problem students consider the relationship between them.
How many different differences can you make?
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
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.
By selecting digits for an addition grid, what targets can you make?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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?
This problem challenges students to work systematically while applying their knowledge of areas of rectangles.
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.
This problem is a good activity for the visualisation of symmetry, and for encouraging students to work systematically.
This is an engaging context in which to reinforce rules of divisibility and challenge students to reason mathematically and work systematically.
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.
This problem offers a simple context for students to explore, make generalisations and prove conjectures, working numerically and algebraically.
This problem is inaccessible without looking at simpler cases, and thus helps students to see the value of specialising in order to generalise.
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?
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?
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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
A collection of short Stage 3 and 4 problems on Working Systematically.
This problem could be used to reinforce work on recording and describing linear sequences.
Can you do a little mathematical detective work to figure out which number has been wiped out?
This problem offers an engaging context in which to apply knowledge of factors, multiples and prime factor decomposition.
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
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?
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
There is nothing half so much worth doing as simply messing about in boats...
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.
As well as introducing the difference of two squares, this problem allows students to explore, conjecture and use algebra to justify their results.
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.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Use the differences to find the solution to this Sudoku.
Can you create a Latin Square from multiples of a six digit number?