Explaining, convincing and proving

problem

Clocked

Age

11 to 14

Challenge level

2 out of 3

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
problem

A Leg to Stand On

Age

11 to 14

Challenge level

2 out of 3

Can you work out the number of chairs at a cafe from the number of legs?
problem

Total Totality

Age

11 to 14

Challenge level

2 out of 3

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?
problem

So Many Sums

Age

11 to 14

Challenge level

2 out of 3

In this addition each letter stands for a different digit, with S standing for 3. What is the value of YxO?
problem

Magical Products

Age

11 to 14

Challenge level

2 out of 3

Can you place the nine cards onto a 3×3 grid such that every row, column and diagonal has a product of 1?
problem

Mini Cross-Number

Age

11 to 14

Challenge level

2 out of 3

Which digit replaces x in this crossnumber?
problem

Digital Book

Age

11 to 14

Challenge level

2 out of 3

If it takes 852 digits to number all the pages of a book, what is the number of the last page?
problem

Sevens

Age

11 to 14

Challenge level

2 out of 3

What is the largest number Sophie can use to have seven positive integers with a mean of 7?
problem

Bookshop

Age

11 to 14

Challenge level

2 out of 3

If Clara spends £23 on books and magazines, how many of each does she buy?
problem

Ones, Twos and Threes

Age

11 to 14

Challenge level

2 out of 3

Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?