List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

Can you find any two-digit numbers that satisfy all of these statements?

Are these statements always true, sometimes true or never true?

Got It game for an adult and child. How can you play so that you know you will always win?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Can you find any perfect numbers? Read this article to find out more...

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...

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

Is there an efficient way to work out how many factors a large number has?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Can you explain the strategy for winning this game with any target?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

Can you find different ways of creating paths using these paving slabs?

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

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

This article for teachers describes how number arrays can be a useful representation for many number concepts.

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

Can you find a way to identify times tables after they have been shifted up or down?

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

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

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

Number problems at primary level that may require resilience.

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

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

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.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

Find the highest power of 11 that will divide into 1000! exactly.

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?"

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

Can you work out some different ways to balance this equation?