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 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?
Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.
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.
A Sudoku based on clues that give the differences between adjacent cells.
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.
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
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?
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?
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?
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?
A Sudoku with a twist.
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 Sudoku with clues as ratios.
Given the products of diagonally opposite cells - can you complete this Sudoku?
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
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?
You need to find the values of the stars before you can apply normal Sudoku rules.
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.
Find out about Magic Squares in this article written for students. Why are they magic?!
A Sudoku with a twist.
A Sudoku with clues as ratios.
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?
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?
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?
Solve the equations to identify the clue numbers in this Sudoku problem.
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
How many different symmetrical shapes can you make by shading triangles or squares?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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.
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.
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?
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?
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?
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?
An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
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.
Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.
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.
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".
In this article, the NRICH team describe the process of selecting solutions for publication on the site.