# Reasoning, Justifying, Convincing and Proof - Upper Secondary

### What's it Worth?

##### Age 11 to 16 Challenge Level:

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

### Take Three from Five

##### Age 14 to 16 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?

### What's Possible?

##### Age 14 to 16 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

##### Age 16 to 18 Challenge Level:

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

### Prime Sequences

##### Age 16 to 18 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

##### Age 16 to 18 Challenge Level:

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

### Polynomial Interpolation

##### Age 16 to 18 Challenge Level:

Can you fit polynomials through these points?

### What's a Group?

##### Age 16 to 18 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?

##### Age 16 to 18 Challenge Level:

Can you make a square from these triangles?

### Impossible Triangles?

##### Age 16 to 18 Challenge Level:

Which of these triangular jigsaws are impossible to finish?

### Dodgy Proofs

##### Age 16 to 18 Challenge Level:

These proofs are wrong. Can you see why?

##### Age 11 to 16 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?

### More Number Sandwiches

##### Age 11 to 16 Challenge Level:

When is it impossible to make number sandwiches?

### Digital Book

##### Age 11 to 16 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?

### Digital Counter

##### Age 11 to 16 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?

### Distinct in a Line

##### Age 11 to 16 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?

### Age Old Lies

##### Age 11 to 16 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?

### Anti-magic Square

##### Age 11 to 16 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...

### Self-referential

##### Age 11 to 16 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?

### Long List

##### Age 14 to 16 Short Challenge Level:

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

### Latin Numbers

##### Age 14 to 16 Challenge Level:

Can you create a Latin Square from multiples of a six digit number?

##### Age 14 to 16 Challenge Level:

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

### Attractive Tablecloths

##### Age 14 to 16 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?

### Arithmagons

##### Age 14 to 16 Challenge Level:

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