Working Systematically - Lower Secondary

Working Systematically is part of our Thinking Mathematically collection.
Scroll down to see our complete collection of problems that require you to work systematically, or explore the two sub-collections focussing on important aspects of systematic working.
Stage: 3

Noticing Patterns

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.

Stage: 3

Finding All Solutions

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!"




We're All Going on a Quadrilateral Hunt

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 38 - 2013
A regular hexagon is divided into six equilateral triangles. How many quadrilaterals are there in the diagram?

Island Hopping

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 49 - 2009
What is the smallest number of ferry trips that Neda needs to take to visit all four islands and return to the mainland?

Satnav Dilemma

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 8 - 2012
How many routes are there in this diagram from S to T?

Two and Two

KS 2 & 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Negative Dice

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 16 - 2012
If the odd numbers on two dice are made negative, which of the totals cannot be achieved?

Jack of Cards

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 32 - 2013
What order did Jack have the cards in to make his card trick work?

Flag-tastic

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 2 - 2007
How many different flags can you make?

Patchwork Quilt

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 33 - 2013
Squares of the type shown are sewn together to make a quilt. How many different quilts can be made?

Famous Five

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 22 - 2007
The Famous Five have been given 20 sweets as a reward for solving a tricky crime.... how many different ways can they share the sweets?

Alberta's Age

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 26 - 2011
Alberta won't reveal her age. Can you work it out from these clues?

Summing Consecutive Numbers

KS 3 Challenge Level:

Challenge Level:1

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?

Shifting Times Tables

KS 3 Challenge Level:

Challenge Level:1

Can you find a way to identify times tables after they have been shifted up?

Double with 1-9

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 31 - 2011
Can you find a number and its double using the digits $1$ to $9$ only once each?

Isometric Rhombi

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 31 - 2016
The diagram shows a grid of $16$ identical equilateral triangles. How many rhombuses are there made up of two adjacent small triangles?

Central Sum

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 53 - 2011
Find a number between 100 and 999 that has its middle digit equal to the sum of the other two digits. Can you find all possibilities?

ACE, TWO, THREE...

KS 3 Challenge Level:

Challenge Level:1

Can you picture how to order the cards to reproduce Charlie's card trick for yourself?

Threes and Fours

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 32 - 2016
What is the smallest integer which has every digit a 3 or a 4 and is divisible by both 3 and 4?

Gridlock

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 29 - 2006
No letter is repeated in any row, column or diagonal. Which letter is in the square marked with the star?

Many Clues, One Answer

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 35 - 2011
You are given lots of clues about a number. Can you work out what it is?

Half and Half

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 44 - 2016
Two of the four small triangles are to be painted black. In how many ways can this be done?

Quick Ticket

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 38 - 2006
What is the fewest coins that must change hands if I buy a ticket for 44p?

Loose Change

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 10 - 2009
In how many ways can you give change for a ten pence piece?

Grid Without Lines

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 40 - 2010
Can you remove the least number of points from this diagram, so no three of the remaining points are in a straight line?

When Shall We Three Meet Again?

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 4 - 2013
On my clock's display, the time has just changed to 02:31. How many minutes will it be until all the digits 0, 1, 2, 3 next appear together again?

Latin Multiplication

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 28 - 2015
Can you choose one number from each row and column in this grid to form the largest possibe product?

American Billions

KS 3 Challenge Level:

Challenge Level:1

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...

Different Digital Clock

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 14 - 2013
At how many times between 10 and 11 o'clock are all six digits on a digital clock different?

Glovely

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 39 - 2006
What is the fewest gloves that Dilly needs to bring Granny to ensure that they have a pair?

M, M and M

KS 3 Challenge Level:

Challenge Level:1

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Kept Apart

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 43 - 2010
The squares of this grid contain one of the letters P, Q, R and S. Can you complete this square so that touching squares do not contain the same letter? How many possibilities are there?

1 Step 2 Step

KS 3 Challenge Level:

Challenge Level:1

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?

Isosceles Triangles

KS 3 Challenge Level:

Challenge Level:1

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?

Making 11p

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 20 - 2006
How many ways are there to make 11p using 1p, 2p and 5p coins?

Musical Chairs

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 42 - 2008
How many different ways could we have sat on the two remaining musical chairs at Gill's fourth birthday party?

Pick's Theorem

KS 3 Challenge Level:

Challenge Level:1

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 They Be Equal?

KS 3 Challenge Level:

Challenge Level:1

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Colourful Tiles

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 21 - 2011
How many ways can you paint this wall with four different colours?

Shady Symmetry

KS 3 Challenge Level:

Challenge Level:1

How many different symmetrical shapes can you make by shading triangles or squares?

Sticky Numbers

KS 3 Challenge Level:

Challenge Level:1

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

End of a Prime

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 25 - 2016
A list is made of every digit that is the units digit of at least one prime number. How many digits appear in the list?

Peaches Today, Peaches Tomorrow....

KS 3 & 4 Challenge Level:

Challenge Level:2 Challenge Level:2

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

Medal Ceremony

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 41 - 2013
The teacher has forgotten which pupil won which medal. In how many different ways could he give the medals out to the pupils?

Even Squares

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 27 - 2008
In the diagram in the question, how many squares, of any size, are there whose entries add up to an even total?

Spead Out

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 24 - 2015
In how many ways can you move through the grid to give the digits 2009?

Rolling Along the Trail

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 32 - 2011
What could be the scores from five throws of this dice?

Phone Call

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 25 - 2015
How many different phone numbers are there starting with a 3 and with at most two different digits?

Nine Colours

KS 3 & 4 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

Charlie's Delightful Machine

KS 3 & 4 Challenge Level:

Challenge Level:1

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Gridlines

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 27 - 2015
How many triples of points are there in this 4x4 array that lie on a straight line?

Domino Hexagon

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 18 - 2017
Dominic wants to place the six dominoes above in a hexagonal ring. Which of the dominoes could be placed next to the one shown?

Amazing

KS 3 & 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 6 - 2014
A maze has nine rooms, with gaps in the walls between them. How many ways are there to travel from X to Y?

Switch On

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 39 - 2015
In how many different ways can a row of five "on/off" switches be set so that no two adjacent switches are in the "off" position?

No Square Sums

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 12 - 2011
How many numbers do you need to remove to avoid making a perfect square?

Fiction in Wonderland

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 17 - 2014
Tweedledum, Tweedledee, Alice and the White Rabbit are having a conversation. How many of the statements they make are true?

Wedding Morning

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 45 - 2015
If Sam is getting married on the 9th of November 2015 aged 30, do you know which year he was born in?

Circle of Lies

KS 3 & 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 34 - 2014
Can you work out how many of Pierre, Qadr, Ratna, Sven and Tanya are telling the truth?

Middle Digit Mean

KS 3 & 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 16 - 2016
How many three digit numbers have the property that the middle digit is the mean of the other two digits?

Mini-sodoku

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 8 - 2015
How many ways are there of completing the mini-sudoku shown?

What's Possible?

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

A Long Time at the Till

KS 4 & 5 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?