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Arithmagons

Stage: 4 Challenge Level:

Can you find the values at the vertices when you know the values on the edges?

Multiplication Arithmagons

Stage: 4 Challenge Level:

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Games Related to Nim

Stage: 1, 2, 3 and 4

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.

Multiplication Square

Stage: 4 Challenge Level:

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Searching for Mean(ing)

Stage: 3 Challenge Level:

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

Sliding Puzzle

Stage: 1, 2, 3 and 4 Challenge Level:

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.

What's Possible?

Stage: 4 Challenge Level:

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Pareq Calc

Stage: 4 Challenge Level:

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

Pair Products

Stage: 4 Challenge Level:

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Jam

Stage: 4 Challenge Level:

A game for 2 players

Sums of Pairs

Stage: 3 and 4 Challenge Level:

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?”

Hypotenuse Lattice Points

Stage: 4 Challenge Level:

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?

Mystic Rose

Stage: 4 Challenge Level:

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Polycircles

Stage: 4 Challenge Level:

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Steel Cables

Stage: 4 Challenge Level:

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 Challenge Level:

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Beelines

Stage: 4 Challenge Level:

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Summing Consecutive Numbers

Stage: 3 Challenge Level:

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?

Where Can We Visit?

Stage: 3 Challenge Level:

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?

Take Three from Five

Stage: 4 Challenge Level:

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Square Pizza

Stage: 4 Challenge Level:

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Generating Triples

Stage: 4 Challenge Level:

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

More Number Pyramids

Stage: 3 Challenge Level:

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Loopy

Stage: 4 Challenge Level:

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Areas of Parallelograms

Stage: 4 Challenge Level:

Can you find the area of a parallelogram defined by two vectors?

Make 37

Stage: 2 and 3 Challenge Level:

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.

Partly Painted Cube

Stage: 4 Challenge Level:

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

In a Spin

Stage: 4 Challenge Level:

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Winning Lines

Stage: 2, 3 and 4

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Stage: 3 Challenge Level:

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?

AMGM

Stage: 4 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

What Numbers Can We Make Now?

Stage: 3 Challenge Level:

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Number Pyramids

Stage: 3 Challenge Level:

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Nim

Stage: 4 Challenge Level:

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.

Cubes Within Cubes Revisited

Stage: 3 Challenge Level:

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?

Nim-like Games

Stage: 2, 3 and 4 Challenge Level:

A collection of games on the NIM theme

Nim-interactive

Stage: 3 and 4 Challenge Level:

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.

One, Three, Five, Seven

Stage: 3 and 4 Challenge Level:

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.

Of All the Areas

Stage: 4 Challenge Level:

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Equilateral Areas

Stage: 4 Challenge Level:

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Magic Squares II

Stage: 4 and 5

An article which gives an account of some properties of magic squares.

Magic Squares

Stage: 4 and 5

An account of some magic squares and their properties and and how to construct them for yourself.

How Much Can We Spend?

Stage: 3 Challenge Level:

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Plus Minus

Stage: 4 Challenge Level:

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Building Gnomons

Stage: 4 Challenge Level:

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

Christmas Chocolates

Stage: 3 Challenge Level:

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Gnomon Dimensions

Stage: 4 Challenge Level:

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

Consecutive Negative Numbers

Stage: 3 Challenge Level:

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

Jam

Stage: 4 Challenge Level:

To avoid losing think of another very well known game where the patterns of play are similar.

Janine's Conjecture

Stage: 4 Challenge Level:

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .