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For younger learners

  • Early Years Foundation Stage

Secondary Summer Challenges 2023

We have a challenge a day for you throughout the summer break...

Each weekday, from 10 July to 1 September, a new problem or game will appear on this page.

After you've had a go at the day's challenge you may be able to compare your approach to the solutions we have published, which are based on students' work. 

You can also find Primary Summer Challenges 2023.

Last Biscuit

Age 11 to 18
Challenge Level Yellow starYellow star

Can you find a strategy that ensures you get to take the last biscuit in this game?

Multiples Sudoku

Age 11 to 14
Challenge Level Yellow star

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

Why 24?

Age 14 to 16
Challenge Level Yellow starYellow star

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Sociable Cards

Age 11 to 14
Challenge Level Yellow star

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

Magic Letters

Age 11 to 14
Challenge Level Yellow star

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

Route to Infinity

Age 11 to 14
Challenge Level Yellow starYellow star

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

Kite in a Square

Age 14 to 18
Challenge Level Yellow starYellow star

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Substitution Cipher

Age 11 to 14
Challenge Level Yellow star

Find the frequency distribution for ordinary English, and use it to help you crack the code.

Cinema Problem

Age 11 to 14
Challenge Level Yellow star

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

Olympic Measures

Age 11 to 14
Challenge Level Yellow star

These Olympic quantities have been jumbled up! Can you put them back together again?

Consecutive Seven

Age 11 to 14
Challenge Level Yellow star

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

More Less Is MoreLive

Age 7 to 14
Challenge Level Yellow star

In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.

Square It

Age 11 to 16
Challenge Level Yellow star

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Special Numbers

Age 11 to 14
Challenge Level Yellow star

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

What Numbers Can We Make?

Age 11 to 14
Challenge Level Yellow star

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

Forwards Add Backwards

Age 11 to 14
Challenge Level Yellow star

What happens when you add a three digit number to its reverse?

What's Possible?

Age 14 to 16
Challenge Level Yellow starYellow star

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

Take Three from Five

Age 11 to 16
Challenge Level Yellow starYellow star

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

Where Can We Visit?

Age 11 to 14
Challenge Level Yellow starYellow star

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Connect Three

Age 11 to 16
Challenge Level Yellow starYellow star

In this game the winner is the first to complete a row of three. Are some squares easier to land on than others?

Got It

Age 7 to 14
Challenge Level Yellow starYellow star

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Frogs

Age 11 to 14
Challenge Level Yellow star

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Pair Products

Age 14 to 16
Challenge Level Yellow star

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Crossing the Bridge

Age 11 to 18
Challenge Level Yellow starYellow star

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

Product Sudoku

Age 11 to 16
Challenge Level Yellow starYellow star

The clues for this Sudoku are the product of the numbers in adjacent squares.

Add to 200

Age 11 to 14
Challenge Level Yellow star

By selecting digits for an addition grid, what targets can you make?

American Billions

Age 11 to 14
Challenge Level Yellow star

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Sticky Numbers

Age 11 to 14
Challenge Level Yellow star

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Overlaps

Age 11 to 14
Challenge Level Yellow starYellow star

Can you find ways to put numbers in the overlaps so the rings have equal totals?

Peaches Today, Peaches Tomorrow...

Age 11 to 14
Challenge Level Yellow starYellow star

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

Factors and Multiples Game

Age 7 to 16
Challenge Level Yellow star

A game in which players take it in turns to choose a number. Can you block your opponent?

Gabriel's Problem

Age 11 to 14
Challenge Level Yellow star

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Cryptarithms

Age 11 to 14
Challenge Level Yellow star

Can you crack these cryptarithms?

Tourism

Age 11 to 14
Challenge Level Yellow starYellow starYellow star

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Always a Multiple?

Age 11 to 14
Challenge Level Yellow starYellow star

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Dozens

Age 7 to 14
Challenge Level Yellow star

Can you select the missing digit(s) to find the largest multiple?

The Remainders Game

Age 7 to 14
Challenge Level Yellow star

Play this game and see if you can figure out the computer's chosen number.

Factors and Multiples Puzzle

Age 11 to 14
Challenge Level Yellow star

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

Getting Round the City

Age 11 to 16

In a city with a grid system of roads, how do you get from A to B?

Zin Obelisk

Age 11 to 14
Challenge Level Yellow starYellow star

In the ancient city of Atlantis a solid rectangular object called a Zin was built in honour of the goddess Tina. Your task is to determine on which day of the week the obelisk was completed.

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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