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.

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

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?

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

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?

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.

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

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?

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

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.

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

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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?

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

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

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

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

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?

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

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

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?

A Sudoku that uses transformations as supporting clues.

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.

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

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".

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.

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

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

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?

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.

This Sudoku, based on differences. Using the one clue number can you find the solution?

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

A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?

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.

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

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?