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

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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.

problem

### What's Possible?

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

problem

### Why 24?

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.

problem

### Where can we visit?

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?

problem

### American Billions

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...

problem

### Frogs

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

problem

### Cinema Problem

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.

problem

### Take Three From Five

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?

problem

### Special Numbers

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?

problem

### Pair Products

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

problem

### Peaches today, Peaches tomorrow...

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?

problem

### Tourism

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.

problem

### Square It

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.

problem

### Consecutive Seven

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?

problem

### Product Sudoku

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

problem

### Substitution Cipher

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

problem

### Factors and Multiples Puzzle

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

problem

### Route to infinity

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

problem

### Connect Three

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

problem

### crossing the bridge

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

problem

### Zin Obelisk

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.

problem

### Multiples Sudoku

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

problem

### Sticky Numbers

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

problem

### Always a multiple?

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

problem

### Sociable Cards

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

problem

### What numbers can we make?

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

problem

### Magic Letters

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

problem

### Kite in a Square

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

problem

### Olympic Measures

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

problem

### Gabriel's Problem

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

problem

### More Less is More

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

problem

### Got It

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.