Or search by topic
There are 212 NRICH Mathematical resources connected to Exploring and noticing, you may find related items under Thinking mathematically.
Broad Topics > Thinking mathematically > Exploring and noticingPlace the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Can you find the values at the vertices when you know the values on the edges?
Can you work out what step size to take to ensure you visit all the dots on the circle?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
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?
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?
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?
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?
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.
Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?
Which set of numbers that add to 100 have the largest product?
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.
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
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.
Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.
The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.
Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?
Amy has a box containing domino pieces but she does not think it is a complete set. Which of her domino pieces are missing?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Find at least one way to put in some operation signs to make these digits come to 100.
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?
During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?
Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number.
There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
There are six numbers written in five different scripts. Can you sort out which is which?
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.
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
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?
Can you explain the strategy for winning this game with any target?
Use these four dominoes to make a square that has the same number of dots on each side.
Use the 'double-3 down' dominoes to make a square so that each side has eight dots.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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.
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
How many faces can you see when you arrange these three cubes in different ways?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.