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?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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?
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?
The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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?
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.
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.
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 numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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?
How many different symmetrical shapes can you make by shading triangles or squares?
Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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 few extra challenges set by some young NRICH members.
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
Four friends must cross a bridge. How can they all cross it in just 17 minutes?
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?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
You need to find the values of the stars before you can apply normal Sudoku rules.
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...
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
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.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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.
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?
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.
Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Find the values of the nine letters in the sum: FOOT + BALL = GAME
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Use the differences to find the solution to this Sudoku.
The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
A Sudoku that uses transformations as supporting clues.
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.
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?
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?
Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?
Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.
Given the products of adjacent cells, can you complete this Sudoku?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
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.
Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?