# Reasoning, Justifying, Convincing and Proof - Upper Secondary

### What's it Worth?

##### Stage: 3 and 4 Challenge Level:

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

### Odds and Evens

##### Stage: 3 and 4 Challenge Level:

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

### Marbles in a Box

##### Stage: 3 and 4 Challenge Level:

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

### Take Three from Five

##### Stage: 3 and 4 Challenge Level:

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?

### Seven Squares

##### Stage: 3 and 4 Challenge Level:

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

### What's Possible?

##### Stage: 4 Challenge Level:

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

### Stats Statements

##### Stage: 5 Challenge Level:

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

### Prime Sequences

##### Stage: 5 Challenge Level:

This group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?

### Integration Matcher

##### Stage: 5 Challenge Level:

Match the charts of these functions to the charts of their integrals.

### Polynomial Interpolation

##### Stage: 5 Challenge Level:

Can you fit polynomials through these points?

### What's a Group?

##### Stage: 5 Challenge Level:

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.

### Impossible Square?

##### Stage: 5 Challenge Level:

Can you make a square from these triangles?

### Impossible Triangles?

##### Stage: 5 Challenge Level:

Which of these triangular jigsaws are impossible to finish?

### Dodgy Proofs

##### Stage: 5 Challenge Level:

These proofs are wrong. Can you see why?

### Route to Infinity

##### Stage: 3 and 4 Challenge Level:

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

### Self-referential

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

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?

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

Weekly Problem 41 - 2007
The Queen of Spades always lies for the whole day or tells the truth for the whole day. Which of these statements can she never say?

### Knights and Knaves

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

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

### Other Side

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

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:

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

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

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

### Mini Cross-number

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

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

### Bookshop

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

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?

### Anti-magic Square

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

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...

### Distinct in a Line

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

Weekly Problem 51 - 2008
This grid can be filled up using only the numbers 1, 2, 3, 4, 5 so that each number appears just once in each row, once in each column and once in each diagonal. Which number goes in the centre square?

### Digital Book

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

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?

### Reasoning, Justifying, Convincing and Proof - Short Problems

##### Stage: 3 and 4

A collection of short Stage 3 and 4 problems on Reasoning, Justifying, Convincing and Proof.

### Digital Counter

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

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

### Reasoning, Justifying, Convincing and Proof - Short Problems

##### Stage: 3 and 4

A collection of short Stage 3 and 4 problems on Reasoning, Justifying, Convincing and Proof.

### Age Old Lies

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

Weekly Problem 20 - 2015
Four brothers give statements about the order they were born in. Can you work out which two are telling the truth?

### Attractive Tablecloths

##### Stage: 4 Challenge Level:

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?