Can you work out how to win this game of Nim? Does it matter if you go first or second?
Can you explain the strategy for winning this game with any target?
A game for 2 players with similaritlies 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.
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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
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.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
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?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
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. . . .
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Can you find sets of sloping lines that enclose a square?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
It starts quite simple but great opportunities for number discoveries and patterns!
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.
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 triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
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
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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.
A collection of games on the NIM theme
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
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 winner is the player to take the last counter.
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Can you explain how this card trick works?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
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?