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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.

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When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

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Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.

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This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

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

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

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?

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

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Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?

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Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. . . .

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There are two forms of counting on Vuvv - Zios count in base 3 and Zepts count in base 7. One day four of these creatures, two Zios and two Zepts, sat on the summit of a hill to count the legs of. . . .

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

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

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

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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.

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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)?

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

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

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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.

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

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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?

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

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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?

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

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

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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?

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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?

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

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

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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. . . .

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

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

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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.

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

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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" .

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How many six digit numbers are there which DO NOT contain a 5?

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Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

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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. . . .

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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.

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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.

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

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

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There are nasty versions of this dice game but we'll start with the nice ones...

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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). . . .

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Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent.

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

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Explore the relationship between simple linear functions and their graphs.