Reasoning, Justifying, Convincing and Proof - Lower Secondary

Reasoning, Justifying, Convincing and Proof is part of our Thinking Mathematically collection.

Summing Consecutive Numbers

Stage: 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?

What's Possible?

Stage: 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?

Marbles in a Box

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

Attractive Tablecloths

Stage: 4 Challenge Level: Challenge Level:1

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

1 Step 2 Step

Stage: 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?

What's it Worth?

Stage: 3 and 4 Challenge Level: Challenge Level:1

There are lots of different methods to find out what the shapes are worth - how many can you find?

Take Three from Five

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Anti-magic Square

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 44 - 2011
You have already used Magic Squares, now meet an Anti-Magic Square. Its properties are slightly different, but can you still solve it...

Number Pyramids

Stage: 3 Challenge Level: Challenge Level:1

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Tilted Squares

Stage: 3 Challenge Level: Challenge Level:1

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Painted Cube

Stage: 3 Challenge Level: Challenge Level:1

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

A Leg to Stand On

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 30 - 2012
Can you work out the number of chairs at a cafe from the number of legs?

Arithmagons

Stage: 3 Challenge Level: Challenge Level:1

Can you find the values at the vertices when you know the values on the edges?

T-table

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 24 - 2013
What is the maximum number of T shaped pieces that can be placed on the grid without overlapping?

More Total Totality

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 26 - 2013
Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?

Sum Total

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 50 - 2013
Each letter in this sum represents a different digit. How many solutions are there?

Odds and Evens

Stage: 3 and 4 Challenge Level: Challenge Level:1

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

Route to Infinity

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you describe this route to infinity? Where will the arrows take you next?

Diminishing Returns

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

Square LCM

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 7 - 2010
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Knights and Knaves

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 35 - 2010
Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?

Takeaway Time

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 27 - 2011
Pizza, Indian or Chinese takeaway. Each teenager from a class only likes two of these, but can you work which two?

What Numbers Can We Make?

Stage: 3 Challenge Level: Challenge Level:1

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Magic Letters

Stage: 3 Challenge Level: Challenge Level:1

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Seven Squares

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Mini Cross-number

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 47 - 2014
Which digit replaces x in this crossnumber?

Digital Book

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 11 - 2015
If it takes 852 digits to number all the pages of a book, what is the number of the last page?

Digital Counter

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 13 - 2015
When the numbers from 1 to 1000 are written on a blackboard, which digit appears the most number of times?

Staircase Sum

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 14 - 2015
The digits 1-9 have been written in the squares so that each row and column sums to 13. What is the value of n?

Weekly Problem 20 - 2015

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Four brothers give statements about the order they were born in. Can you work out which two are telling the truth?

Self-referential

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 30 - 2015
How many ways are there of completing this table so that each row tells you how many there are of the numbers 1, 2, 3 and 4?

Multiplication Magic Square

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 32 - 2015
Can you work out the missing numbers in this multiplication magic square?

Down and Along

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 50 - 2015
Can you work out the values of J, M and C in this sum?

Other Side

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 8 - 2016
The diagram shows a quadrilateral $ABCD$, in which $AD=BC$, $\angle CAD=50^\circ$, $\angle ACD=65^\circ$ and $\angle ACB=70^\circ$. What is the size of $\angle ABC$?

Equilateral Pair

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 39 - 2016
In the diagram, VWX and XYZ are congruent equilateral triangles. What is the size of angle VWY?

Shaded Square

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 41 - 2016
The diagram shows a square, with lines drawn from its centre. What is the shaded area?

Peter's Primes

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 22 - 2017
Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?

Bookshop

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 3 - 2017
Books cost £3.40 and magazines cost £1.60. If Clara spends £23 on books and Magazines, how many of each does she buy?

Shared Vertex

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 38 - 2017
In the diagram, what is the value of $x$?

Long List

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?