This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

Given the products of diagonally opposite cells - can you complete this Sudoku?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Two sudokus in one. Challenge yourself to make the necessary connections.

Each clue number in this sudoku is the product of the two numbers in adjacent cells.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

Given the products of adjacent cells, can you complete this Sudoku?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

Can you use this information to work out Charlie's house number?

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Two sudokus in one. Challenge yourself to make the necessary connections.

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?