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

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

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?

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?

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?

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?

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

A few extra challenges set by some young NRICH members.

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?

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.

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?

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?

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?

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

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

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?

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?

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

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.

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

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

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

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

Use the differences to find the solution to this Sudoku.

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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

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

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

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. . . .

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

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.

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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

A Sudoku that uses transformations as supporting clues.

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

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

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

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?

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

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

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.

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

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.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

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