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There are 319 NRICH Mathematical resources connected to Conjecturing and generalising, you may find related items under Thinking mathematically.
Broad Topics > Thinking mathematically > Conjecturing and generalisingExplore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
In how many different ways can you break up a stick of seven interlocking cubes? Now try with a stick of eight cubes and a stick of six cubes. What do you notice?
One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
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?
Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
There are lots of ideas to explore in these sequences of ordered fractions.
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
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 know?
Four bags contain a large number of 1s, 3s, 5s and 7s. Can you pick any ten numbers from the bags so that their total is 37?
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.
Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
Which set of numbers that add to 100 have the largest product?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
Amy has a box containing domino pieces but she does not think it is a complete set. Which of her domino pieces are missing?
On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?
What do you notice about these squares of numbers? What is the same? What is different?
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?
There are six numbers written in five different scripts. Can you sort out which is which?
Can you find sets of sloping lines that enclose a square?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
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.
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?
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
Is there an efficient way to work out how many factors a large number has?
Can you explain the strategy for winning this game with any target?
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.