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Broad Topics > Numbers and the Number System > Properties of numbers

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Really Mr. Bond

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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Rachel's Problem

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!

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Enriching Experience

Stage: 4 Challenge Level: Challenge Level:1

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

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Filling the Gaps

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Which numbers can we write as a sum of square numbers?

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Factorial

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

How many zeros are there at the end of the number which is the product of first hundred positive integers?

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Pair Products

Stage: 4 Challenge Level: Challenge Level:1

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

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Even So

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Lastly - Well

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What are the last two digits of 2^(2^2003)?

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Times Right

Stage: 3 and 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level: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?

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Robert's Spreadsheet

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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Elevenses

Stage: 3 Challenge Level: Challenge Level:1

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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A Long Time at the Till

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

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Guess the Dominoes

Stage: 1, 2 and 3 Challenge Level: Challenge Level:1

This task depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together.

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Sept 03

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

What is the last digit of the number 1 / 5^903 ?

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Helen's Conjecture

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

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Writ Large

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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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Arrange the Digits

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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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Generating Triples

Stage: 4 Challenge Level: Challenge Level:1

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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Slippy Numbers

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The number 10112359550561797752808988764044943820224719 is called a 'slippy number' because, when the last digit 9 is moved to the front, the new number produced is the slippy number multiplied by 9.

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Four Coloured Lights

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

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Fracmax

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.

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A Little Light Thinking

Stage: 4 Challenge Level: Challenge Level:1

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

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Cogs

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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Palindromes

Stage: 1, 2 and 3

Find out about palindromic numbers by reading this article.

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Six Times Five

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Odd Differences

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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Happy Octopus

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This investigation is about happy numbers in the World of the Octopus where all numbers are written in base 8 ... Find all the fixed points and cycles for the happy number sequences in base 8.

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Whole Numbers Only

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you work out how many of each kind of pencil this student bought?

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Mini-max

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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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The Patent Solution

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

A combination mechanism for a safe comprises thirty-two tumblers numbered from one to thirty-two in such a way that the numbers in each wheel total 132... Could you open the safe?

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Lesser Digits

Stage: 3 Challenge Level: Challenge Level:1

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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Chameleons

Stage: 3 Challenge Level: Challenge Level:1

Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .

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Unit Fractions

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

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Counting Factors

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Not a Polite Question

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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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Can You Find a Perfect Number?

Stage: 2 and 3

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

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Clever Carl

Stage: 2 and 3

What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.

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Got it Article

Stage: 2 and 3

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

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The Codabar Check

Stage: 3

This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.

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Water Lilies

Stage: 3 Challenge Level: Challenge Level:1

There are some water lilies in a lake. The area that they cover doubles in size every day. After 17 days the whole lake is covered. How long did it take them to cover half the lake?

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Alphabet Soup

Stage: 3 Challenge Level: Challenge Level:1

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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Please Explain

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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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X Marks the Spot

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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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Two Much

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Thirty Six Exactly

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Oh! Hidden Inside?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Magic Letters

Stage: 3 Challenge Level: Challenge Level:1

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

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Power Crazy

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

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Repetitiously

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

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Small Change

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?


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