Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
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)?
Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?
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?
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. . . .
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Find b where 3723(base 10) = 123(base b).
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?
A job needs three men but in fact six people do it. When it is finished they are all paid the same. How much was paid in total, and much does each man get if the money is shared as Fred suggests?
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?
A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Find the five distinct digits N, R, I, C and H in the following nomogram
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...
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
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?
Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
How to build your own magic squares.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?
Show that all pentagonal numbers are one third of a triangular number.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
Can you find a rule which connects consecutive triangular numbers?
Can you make sense of these three proofs of Pythagoras' Theorem?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?