Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!
Find the five distinct digits N, R, I, C and H in the following
Which numbers can we write as a sum of square numbers?
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?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
What are the last two digits of 2^(2^2003)?
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?
What is the last digit of the number 1 / 5^903 ?
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?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.
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.
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
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?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
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. . . .
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?
Find out about palindromic numbers by reading this article.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
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
How many six digit numbers are there which DO NOT contain a 5?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
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?
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
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.
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.
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. . . .
How many positive integers less than or equal to 4000 can be
written down without using the digits 7, 8 or 9?
Tim Rowland introduces irrational numbers
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?
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
Can you find any perfect numbers? Read this article to find out more...
Is there an efficient way to work out how many factors a large number has?
Can you work out how many of each kind of pencil this student
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?
If you take a three by three square on a 1-10 addition square and
multiply the diagonally opposite numbers together, what is the
difference between these products. Why?
This task depends on learners sharing reasoning, listening to
opinions, reflecting and pulling ideas together.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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.
Work out how to light up the single light. What's the rule?
Take three whole numbers. The differences between them give you
three new numbers. Find the differences between the new numbers and
keep repeating this. What happens?
Robert noticed some interesting patterns when he highlighted square
numbers in a spreadsheet. Can you prove that the patterns will
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?
Can you make a hypothesis to explain these ancient numbers?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nĀ² Use the diagram to show that any odd number is the difference of two squares.
Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?
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" .