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.

problem

Favourite

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?

problem

Favourite

Why 24?

Age

14 to 16

Challenge level

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

Favourite

Where can we visit?

Age

11 to 14

Challenge level

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

Favourite

American billions

Age

11 to 14

Challenge level

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

Favourite

Frogs

Age

11 to 14

Challenge level

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

problem

Favourite

Peaches today, peaches tomorrow...

Age

11 to 14

Challenge level

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

Age

11 to 14

Challenge level

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

Favourite

Consecutive seven

Age

11 to 14

Challenge level

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

Favourite

Substitution cipher

Age

11 to 14

Challenge level

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

problem

Favourite

Factors and multiples puzzle

Age

11 to 14

Challenge level

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

problem

Favourite

Route to infinity

Age

11 to 14

Challenge level

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

problem

Zin obelisk

Age

11 to 14

Challenge level

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

Favourite

Sticky numbers

Age

11 to 14

Challenge level

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

problem

Favourite

Always a multiple?

Age

11 to 14

Challenge level

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

problem

Favourite

Sociable cards

Age

11 to 14

Challenge level

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

problem

Favourite

Magic letters

Age

11 to 14

Challenge level

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

problem

Favourite

Kite in a square

Age

14 to 18

Challenge level

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

problem

Favourite

Olympic measures

Age

11 to 14

Challenge level

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

problem

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

problem

Favourite

Cinema problem

Age

11 to 14

Challenge level

A cinema has 100 seats. How can ticket sales make £100 for these different combinations of ticket prices?

problem

Favourite

Got it

Age

7 to 14

Challenge level

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.

problem

Favourite

Take three from five

Age

11 to 16

Challenge level

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?

game

Favourite

The remainders game

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

game

Last biscuit

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

problem

Favourite

Pair products

Age

14 to 16

Challenge level

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

problem

Favourite

Special numbers

Age

11 to 14

Challenge level

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?