Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
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?
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. . . .
This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.
Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own. . . .
Label this plum tree graph to make it totally magic!
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
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...
Can you find a rule which relates triangular numbers to square numbers?
Show that all pentagonal numbers are one third of a triangular number.
Balance the bar with the three weight on the inside.
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
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?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Make some loops out of regular hexagons. What rules can you discover?
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?
Can you find a rule which connects consecutive triangular numbers?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
How good are you at finding the formula for a number pattern ?
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?
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
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?
The answer is $5x+8y$... What was the question?
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?
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?
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Can you use the diagram to prove the AM-GM inequality?
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. . . .
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?
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.
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?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?
Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .
Where should you start, if you want to finish back where you started?
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?