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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 you to work systematically and will help you appreciate the benefits of working in this way.
Scroll down to see our complete collection of Upper Primary problems, or explore the two sub-collections.
We also have a collection of Lower Primary problems that lend themselves to working systematically.
Primary teachers may like to read our article Encouraging Primary Children to Work Systematically.
These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.
These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?
What could the half time scores have been in these Olympic hockey matches?
How many possible symmetrical necklaces can you find? How do you know you've found them all?
How many different triangles can you make on a circular pegboard that has nine pegs?
Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
Can you use the information to find out which cards I have used?
Amy's mum had given her £2.50 to spend. She bought four times as many pens as pencils and was given 40p change. How many of each did she buy?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
Can you replace the letters with numbers? Is there only one solution in each case?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?
How do you know if your set of dominoes is complete?
Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?
Can you go through this maze so that the numbers you pass add to exactly 100?
Four bags contain a large number of 1s, 3s, 5s and 7s. Can you pick any ten numbers from the bags so that their total is 37?
This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Can you make square numbers by adding two prime numbers together?
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
This challenge combines addition, multiplication, perseverance and even proof.
There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?
This task combines spatial awareness with addition and multiplication.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
This dice train has been made using specific rules. How many different trains can you make?
Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?
A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?
How could you arrange at least two dice in a stack so that the total of the visible spots is 18?
On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
These practical challenges are all about making a 'tray' and covering it with paper.
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.