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

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.

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 is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.

Solve the equations to identify the clue numbers in this Sudoku problem.

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

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?

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?

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.

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

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

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

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

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

A Sudoku that uses transformations as supporting clues.

Given the products of diagonally opposite cells - can you complete this Sudoku?

A few extra challenges set by some young NRICH members.

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

A pair of Sudoku puzzles that together lead to a complete solution.

Find out about Magic Squares in this article written for students. Why are they magic?!

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

Four small numbers give the clue to the contents of the four surrounding cells.

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

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

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?

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

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?

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.

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

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?

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?

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?

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.

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?

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?