Using balancing scales what is the least number of weights needed to weigh all integer masses from 1 to 1000? Placing some of the weights in the same pan as the object how many are needed?

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

This article for the young and old talks about the origins of our number system and the important role zero has to play in it.

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

Find the values of the nine letters in the sum: FOOT + BALL = GAME

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

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

Where should you start, if you want to finish back where you started?

Explore the relationship between simple linear functions and their graphs.

The Number Jumbler can always work out your chosen symbol. Can you work out how?

Try out some calculations. Are you surprised by the results?

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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.

How many solutions can you find to this sum? Each of the different letters stands for a different number.

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

Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Can you create a Latin Square from multiples of a six digit number?

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.

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

Nowadays the calculator is very familiar to many of us. What did people do to save time working out more difficult problems before the calculator existed?

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

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

Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

This challenge is to make up YOUR OWN alphanumeric. Each letter represents a digit and where the same letter appears more than once it must represent the same digit each time.

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

What is the sum of all the digits in all the integers from one to one million?

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

Find the five distinct digits N, R, I, C and H in the following nomogram

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 convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

The number 3723(in base 10) is written as 123 in another base. What is that base?

This article, written for teachers, looks at the different kinds of recordings encountered in Primary Mathematics lessons and the importance of not jumping to conclusions!