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?
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)?
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?
Find the five distinct digits N, R, I, C and H in the following
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
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.
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
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?
Balance the bar with the three weight on the inside.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
Find b where 3723(base 10) = 123(base b).
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Attach weights of 1, 2, 4, and 8 units to the four attachment
points on the bar. Move the bar from side to side until you find a
balance point. Is it possible to predict that position?
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
How to build your own magic squares.
Use algebra to reason why 16 and 32 are impossible to create as the
sum of consecutive numbers.
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?
My train left London between 6 a.m. and 7 a.m. and arrived in Paris
between 9 a.m. and 10 a.m. At the start and end of the journey the
hands on my watch were in exactly the same positions but the. . . .
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. . . .
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?
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
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?
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?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
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?
If the sides of the triangle in the diagram are 3, 4 and 5, what is
the area of the shaded square?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .
A box has faces with areas 3, 12 and 25 square centimetres. What is
the volume of the box?
A hallway floor is tiled and each tile is one foot square. Given
that the number of tiles around the perimeter is EXACTLY half the
total number of tiles, find the possible dimensions of the hallway.
Label this plum tree graph to make it totally magic!
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?
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.
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
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
Think of a number... follow the machine's instructions. I know what
your number is! Can you explain how I know?
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?
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?
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
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?
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
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?”
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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?
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?