Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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. . . .
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?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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. . . .
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.
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?
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?
This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
What is the smallest number with exactly 14 divisors?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
Work out how to light up the single light. What's the rule?
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. . . .
What are the last two digits of 2^(2^2003)?
Can you find any perfect numbers? Read this article to find out more...
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
How many six digit numbers are there which DO NOT contain a 5?
Robert noticed some interesting patterns when he highlighted square
numbers in a spreadsheet. Can you prove that the patterns will
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?
What is the last digit of the number 1 / 5^903 ?
This package contains a collection of problems from the NRICH
website that could be suitable for students who have a good
understanding of Factors and Multiples and who feel ready to take
on some. . . .
Find out about palindromic numbers by reading this article.
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?
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?
A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.
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?
Find the five distinct digits N, R, I, C and H in the following
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
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?
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
Can you make a hypothesis to explain these ancient numbers?
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" .
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?
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.
Which numbers can we write as a sum of square numbers?
Tim Rowland introduces irrational numbers
How many positive integers less than or equal to 4000 can be
written down without using the digits 7, 8 or 9?
Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!
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.
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?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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.
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.
How many zeros are there at the end of the number which is the
product of first hundred positive integers?