The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?
Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.
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.
How many trains can you make which are the same length as Matt's and Katie's, using rods that are identical?
Use the clues to colour each square.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.
A Sudoku with clues given as sums of entries.
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.
Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?
Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
How will you go about finding all the jigsaw pieces that have one peg and one hole?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?
Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?
A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?
What is the best way to shunt these carriages so that each train can continue its journey?
What happens when you try and fit the triomino pieces into these two grids?
Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?
Find out about Magic Squares in this article written for students. Why are they magic?!
Have a go at this game which has been inspired by the Big Internet Math-Off 2019. Can you gain more columns of lily pads than your opponent?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
Here are some rods that are different colours. How could I make a yellow rod using white and red rods?
Investigate the different ways that fifteen schools could have given money in a charity fundraiser.
These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?
Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.
My coat has three buttons. How many ways can you find to do up all the buttons?
Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?
I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?
In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
What could the half time scores have been in these Olympic hockey matches?