Plus Blog

December 7, 2015

Sometimes a piece of mathematics can be so neat and elegant, it makes you want to shout "eureka!" even if you haven't produced it yourself. One of our favourite examples of this is the art gallery problem.

Gallery

The Guggenheim Museum in Bilbao: hard to supervise. Image: MykReeve.

Suppose you have an art gallery containing priceless paintings and sculptures. You would like it to be supervised by security guards, and you want to employ enough of them so that at any one time the guards can between them oversee the whole gallery. How many guards will you need?

Think about this for a while (go on, it's Sunday) and once you've had enough, read about the answer and its proof here. It's pure genius!

This article was inspired by Wild Maths, which encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Return to the Plus Advent Calendar

December 5, 2015

Got it! a game for two players. The first player chooses a whole number from 1 to 4. After that players take turns to add a whole number from 1 to 4 to the running total. The player who hits the target of 23 wins the game.

23

You can play the game against a friend, or against the computer using the interactivity on Wild Maths. Can you find a winning strategy? If yes, can you describe it? And what if you change the target number to something other than 23, or the numbers you are allowed to add to something other than 1 to 4?

Have fun!

Wild Maths encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Return to the Plus Advent Calendar

December 3, 2015

If you want to draw a rhombus on dotty paper, can you start with any two dots?

Rhombuses

Explore the question with the interactivity on the Wild Maths website, where you can also find some follow-up questions. Have fun!

Wild Maths encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Return to the Plus Advent Calendar

December 3, 2015

Can you fold a piece of paper in half? Of course you can, it's easy, you just match the two corners along one side. But can you fold it in thirds? You might be able to with a bit of fiddling and guessing, but what about fifths? Or sevenths? Or thirteenths? There is a simple way you can fold a piece of paper into any fraction you would like – exactly – no guessing or fiddling needed!

To find out how to do it, read Folding fractions.

This article was inspired by content on Wild Maths, which encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Return to the Plus Advent Calendar

December 2, 2015

Sometimes real progress in maths comes when you find a way of looking at a problem in two different ways. Here is a great example of this.

handshake

Suppose you have $n+1$ people in a room and each person shakes hands with each other person once. How many handshakes do you get in total? The first person shakes hands with $n$ other people, the second shakes hands with the $n-1$ remaining people, the third shakes hands with $n-2$ remaining people, etc, giving a total of

$n+(n-1)+(n-2)+...+2+1$ handshakes.

But we can also look at this in another way: each person shakes hands with $n$ others and there are $n+1$ people, giving $n \times (n+1)$ handshakes. But this counts every handshake twice, so we need to divide by 2, giving a total of

  \[ \frac{n \times (n+1)}{2} \]    

handshakes.

Putting these two arguments together, we have just come up with the formula for summing the first $n$ integers and we’ve proved that it is correct:

  \[ n+(n-1)+(n-2)+...+2+1 = \frac{n \times (n+1)}{2}. \]    

This puzzle is inspired by content on our sister site Wild Maths, which encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Return to the Plus Advent Calendar

1 comments
December 1, 2015

Only three things in life are certain: death, taxes and parking fees. But even a menacing parking meter is an excuse to do some maths.

Mini

Imagine, for example, that the car park costs £1.50. The machine only accepts 10p and 20p coins. There are obviously different ways of putting the money into the car park machine, for example

10p, 10p, 20p, 20p, 10p, 10p, 10p, 10p, 20p, 10p, 10p, 10p

or

10p, 10p, 10p, 10p, 20p, 20p, 20p, 10p, 20p, 20p.

You could probably go for the rest of the month without feeding the machine in the same way twice. Can you feed the machine in a different way each day of the year?

You can find a longer version of this puzzle, including some follow-up questions to investigate, on the Wild Maths site. Wild Maths encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Return to the Plus Advent Calendar

Syndicate content