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Working Systematically is part of our Thinking Mathematically collection.
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 you to work systematically, and will help you 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. Organising your work systematically allows you to notice what might not otherwise be obvious.
These problems challenge you 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!"
How many solutions can you find to this sum? Each of the different letters stands for a different number.
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
How many different symmetrical shapes can you make by shading triangles or squares?
Can you find a way to identify times tables after they have been shifted up or down?
A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?
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?
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How many different differences can you make?
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?
By selecting digits for an addition grid, what targets can you make?
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?
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?
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
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?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Can you do a little mathematical detective work to figure out which number has been wiped out?
The clues for this Sudoku are the product of the numbers in adjacent squares.
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
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...
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?