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Chess

Stage: 3 Challenge Level:

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?

Christmas Chocolates

Stage: 3 Challenge Level:

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

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?

Picturing Triangular Numbers

Stage: 3 Challenge Level:

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Tourism

Stage: 3 Challenge Level:

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.

More Magic Potting Sheds

Stage: 3 Challenge Level:

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?

Maths Trails

Stage: 2 and 3

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.

Mirror, Mirror...

Stage: 3 Challenge Level:

Explore the effect of reflecting in two parallel mirror lines.

Cunning Card Trick

Stage: 3 Challenge Level:

Delight your friends with this cunning trick! Can you explain how it works?

Painted Cube

Stage: 3 Challenge Level:

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Squares, Squares and More Squares

Stage: 3 Challenge Level:

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

Threesomes

Stage: 3 Challenge Level:

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Card Trick 2

Stage: 3 Challenge Level:

Can you explain how this card trick works?

Picturing Square Numbers

Stage: 3 Challenge Level:

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Nim-7

Stage: 1, 2 and 3 Challenge Level:

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Is There a Theorem?

Stage: 3 Challenge Level:

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?

2001 Spatial Oddity

Stage: 3 Challenge Level:

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

Squares in Rectangles

Stage: 3 Challenge Level:

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?

Route to Infinity

Stage: 3 Challenge Level:

Can you describe this route to infinity? Where will the arrows take you next?

...on the Wall

Stage: 3 Challenge Level:

Explore the effect of reflecting in two intersecting mirror lines.

Handshakes

Stage: 3 Challenge Level:

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Intersecting Circles

Stage: 3 Challenge Level:

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Mini-max

Stage: 3 Challenge Level:

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

Lower Bound

Stage: 3 Challenge Level:

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Sum Equals Product

Stage: 3 Challenge Level:

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

Steps to the Podium

Stage: 2 and 3 Challenge Level:

It starts quite simple but great opportunities for number discoveries and patterns!

Special Sums and Products

Stage: 3 Challenge Level:

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.

Converging Means

Stage: 3 Challenge Level:

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. . . .

Enclosing Squares

Stage: 3 Challenge Level:

Can you find sets of sloping lines that enclose a square?

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.

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.

More Number Pyramids

Stage: 3 Challenge Level:

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

More Twisting and Turning

Stage: 3 Challenge Level:

It would be nice to have a strategy for disentangling any tangled ropes...

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?

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.

Magic Letters

Stage: 3 Challenge Level:

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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.

For Richer for Poorer

Stage: 3 Challenge Level:

Charlie has moved between countries and the average income of both has increased. How can this be so?

Number Pyramids

Stage: 3 Challenge Level:

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

Repeaters

Stage: 3 Challenge Level:

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.

Partitioning Revisited

Stage: 3 Challenge Level:

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

Egyptian Fractions

Stage: 3 Challenge Level:

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.

Jam

Stage: 4 Challenge Level:

A game for 2 players

Shear Magic

Stage: 3 Challenge Level:

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

All Tangled Up

Stage: 3 Challenge Level:

Can you tangle yourself up and reach any fraction?

Tilted Squares

Stage: 3 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Stage: 3 Challenge Level:

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?

Go Forth and Generalise

Stage: 3

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.

Konigsberg Plus

Stage: 3 Challenge Level:

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.

Three Times Seven

Stage: 3 Challenge Level:

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?