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 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
Can you explain the strategy for winning this game with any target?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
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.
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.
Got It game for an adult and child. How can you play so that you know you will always win?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Investigate the different ways that fifteen schools could have given money in a charity fundraiser.
Can all unit fractions be written as the sum of two unit fractions?
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
Can you find the values at the vertices when you know the values on the edges?
A collection of games on the NIM theme
Delight your friends with this cunning trick! Can you explain how it works?
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
A game for 2 players
To avoid losing think of another very well known game where the patterns of play are similar.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
It starts quite simple but great opportunities for number discoveries and patterns!
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Can you use the diagram to prove the AM-GM inequality?
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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?
It would be nice to have a strategy for disentangling any tangled ropes...
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?