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.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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.

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

How many different symmetrical shapes can you make by shading triangles or squares?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

The clues for this Sudoku are the product of the numbers in adjacent squares.

You need to find the values of the stars before you can apply normal Sudoku rules.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

A Sudoku based on clues that give the differences between adjacent cells.

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?

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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.

Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem.

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?

Two sudokus in one. Challenge yourself to make the necessary connections.

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

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?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

A Sudoku that uses transformations as supporting clues.

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

A few extra challenges set by some young NRICH members.

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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.

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

Use the clues about the shaded areas to help solve this sudoku

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?

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

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?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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?

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?

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

Two sudokus in one. Challenge yourself to make the necessary connections.

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.