Answer to Maths Challenge: Diamond Frame

Follow-up to Maths Challenge: Diamond Frame from IGGY Team blog

Answer to Maths Challenge: Diamond Frame

One of the ten answers: sums are all 18

There are ten basically different answers, where, for example, swapping the ace and seven in the right-hand side doesn’t count as different since it’s a simple transformation that obviously keeps the sums the same. There are two with a sum of 18, four with 19, two with 20, and two with 22.

Here’s one of them.

Karen Ramsay-Smith : 20 Feb 2012 12:06 |  Tags: Challenge Maths |  Comments (0) |  Close comments |  Report a problem

Rapunzel, Leonardo and the Physics of the Ponytail

The famous Grimm’s fairytale of Rapunzel tells the tale of a beautiful girl who is shut away in a tower in the middle of the woods, with neither stairs nor a door, and only one room and one window. On visiting Rapunzel her captor stands beneath the tower and calls out:

Rapunzel, Rapunzel, let down your hair, so that I may climb the golden stair.

The properties of the humble human hair have been of great interest to historical figures including Leonardo Da Vinci and the Brothers Grimm and could now have implications for industry in the areas of textiles, computer animation and personal care products.

Physicists Professor Robin Ball from University of Warwick and Professor Raymond Goldstein from the University of Cambridge have measured the “curliness” of human hair and have produced a mathematical theory that explains the shape of a ponytail.

The research provides understanding of how human hairs are distributed in a ponytail.

The equation takes into account

• stiffness of the hairs
• the effects of gravity on hair
• the presence of the random curliness or waviness of human hair.

Together with a new quantity known as “the Rapunzel Number” – the equation can, they say, be used to predict the shape of any ponytail.

Images from ponytails composing of approximately 10,000 human hairs, each 25 cm long, from commercial hair swatches were analysed mathematically to determine the swelling pressure from the random curvatures of hairs.

The research has important implications for understanding the structure of many materials made up of random fibres, such as wool and fur. The research will also have significance within the computer graphics and animation industry, where the representation of hair has been a challenging problem.

“It’s a remarkably simple equation,” explained Professor Goldstein. “Our findings extend some central paradigms in statistical physics and show how they can be used to solve a problem that has puzzled scientists and artists ever since Leonardo da Vinci remarked on the fluid-like streamlines of hair in his notebooks 500 years ago.

Joanna Thomas : 20 Feb 2012 09:27 |  Tags: Maths Physics Research |  Comments (0) |  Close comments |  Report a problem

Answer to Maths Challenge: Pig on a Rope

Follow-up to Pig on a Rope from IGGY Team blog

Answer to Maths Challenge: Pig on a Rope

To simplify the problem, make six copies of the field, with six copies of the (shaded) region accessible to the pig. Then we want the shaded circle to have half the area of the hexagon.

The area of the circle is where r is the radius. The hexagon has sides 100 metres long, so its area is

and so half of this is

We therefore need

and so

which is about 64.3037 metres.

Karen Ramsay-Smith : 13 Feb 2012 09:07 |  Tags: Challenge Maths |  Comments (0) |  Close comments |  Report a problem

Maths Challenge: Diamond Frame

Maths Challenge: Diamond Frame

Innumeratus had taken the ace to ten of diamonds from a pack of cards, and was arranging them to make a rectangular frame.

‘Look!’ He shouted to Mathophila. ‘I’ve arranged them so that the total number of pips along each side of the frame is the same!’

Mathophila had learned to take such statements with a pinch of salt, and she quickly pointed out that the sums concerned were 19 (top), 20 (left), 22 (right) and 16 (bottom).

‘Well. I’ve arranged them so that the total number of pips along each side of the frame is different, then.’

Mathophila agreed with that, but felt it was a silly puzzle. She really liked the first version better.

Can you solve the original version? You can turn the cards through a right angle if you wish.

Add your answers below, and we'll let you know the solution next week!

Karen Ramsay-Smith : 10 Feb 2012 11:32 |  Tags: Challenge Maths |  Comments (0) |  Close comments |  Trackbacks (1) |  Report a problem

Answer to Maths Challenge: Alien Encounter!

Follow-up to Alien Encounter! from IGGY Team blog

Answer to Maths Challenge: Alien Encounter!

Did you work it out?

Alfy is a Veracitor, whereas Betty and Gemma are Gibberish.

There are only eight possibilities, so you can try each in turn. But there’s a quicker way. Betty said that Alfy and Gemma belong to the same species, but they have given different answers to the same question, so Betty is Gibberish. Alfy said precisely that, making him a Veracitor. Gemma said the opposite, so she must be Gibberish.

Watch the video below

Karen Ramsay-Smith : 06 Feb 2012 08:59 |  Tags: Challenge Maths |  Comments (0) |  Close comments |  Report a problem

Pig on a Rope

Maths Challenge: Pig on a Rope

Farmer Hogswell owns a field, which is a perfect equilateral triangle, each side 100 metres long. His prize pig Pigasus is tied to one corner, so that the portion of the field that Pigasus can reach is exactly half the total area.

How long is the rope?

You may – indeed, must – assume that the pig has zero size (which admittedly is pretty silly) and that the rope is indefinitely thin and any necessary knots can be ignored.

How long do you think the rope is? Please post your ideas below, we'll let you know the correct answer next week!

Karen Ramsay-Smith : 03 Feb 2012 14:19 |  Tags: Challenge Maths |  Comments (6) |  Close comments |  Trackbacks (1) |  Report a problem

Alien Encounter!

Maths Challenge: Alien Encounter!

The starship Indefensible was in orbit around the planet Noncomposmentis, and Captain Quirk and Mr Crock had beamed down to the surface.

According to the Good Galaxy Guide, there are two species of intelligent aliens on this planet.’ said Quirk. ‘
‘Correct, Captain – Veracitors and Gibberish. They all speak Galaxic, and they can be distinguished by how they answer questions.

The Veracitors always reply truthfully, and the Gibberish always lie.

‘But physically-‘
‘_they are indistinguishable, Captain.’

Can you work out which Alien is from which species?

Watch the video below and see if you can work it out!

Answers will be posted next week!

Karen Ramsay-Smith : 30 Jan 2012 09:43 |  Tags: Challenge Maths |  Comments (2) |  Close comments |  Trackbacks (1) |  Report a problem

Answer to Maths Challenge: Dragon Curve

Follow-up to Happy Chinese New Year 2012 – The Year of the Dragon from IGGY Team blog

Answer to Maths Challenge: Dragon Curve

Dragon curves can be made by repeatedly folding a strip of paper in half – always folding the same way - and then opening it out to make all folds into right angles.

These curves determine a fractal. In fact the infinite limit is a space filling curve, but the region it fills has a complicated, dragon-like shape. The sequence of right (R) and left (L) turns in the curve goes like this.

Step 1 R
Step 2 R R L
Step 3 R R L R R L L
Step 4 R R L R R L L R R R L L R L L

In fact, there is a simple pattern: each sequence is formed from the previous one by placing an extra R at the end, followed by the reverse of the previous sequence with R’s and L’s swapped. We have marked the extra R in the middle in bold.

The dragon curve was discovered by John Heighway, Bruce Banks and William Harter – all physicists at NASA – and was mentioned in Martin Gardener’s Mathematical Games’ column in Scientific American in 1967. It has lots of intriguing features – see http://en.wikipedia.org/wiki/Dragon_curve

Karen Ramsay-Smith : 30 Jan 2012 08:58 |  Tags: Challenge Festival Maths |  Comments (0) |  Close comments |  Report a problem

Answer to Maths Challenge: Après–le–Ski

Follow-up to The 2012 Winter Youth Olympics from IGGY Team blog

Answer to Maths Challenge: Après-le-Ski

The cables cross at a height of 240 metres.

It’s simpler to tackle a more general problem, where the lengths are as shown. By similar triangles,

Adding, we get

Dividing by, we obtain

Leading to

We notice that c does not depend on x or y , which is a good job since the puzzle didn’t tell us those. We know that a = 600 , and b = 400 , so

Well done if you got it right!

Karen Ramsay-Smith : 23 Jan 2012 11:28 |  Tags: Challenge Maths Olympics |  Comments (0) |  Close comments |  Report a problem

Happy Chinese New Year 2012 – The Year of the Dragon

Kung Hei Fat Choy! January 23rd marks Chinese New Year 2012. The festival begins with Chinese New Year' Eve on January 22nd and ends on February 6th with the famous Lantern Festival.

2012 is the year of dragon and will be the 4710th Chinese New Year, according to the Chinese zodiac calendar.

The dragon is the 5th sign of the Chinese zodiac and it is regarded as an auspicious symbol which stands for power, good luck, success, and happiness.

To mark the occasion we have a new mini Maths Challenge for you today!

Maths Challenge: Dragon Curve

The picture shows a sequence of curves; called dragon curves (look at the last one).

The sequence can be continued indefinitely, getting ever more complicated curves.

What is the rule for making them?

Ignore the rounding of the corners by the short lines, which is done so that later curves in the sequence remain intelligible.

You can add your ideas in the comments below.

Why not have a go at folding some dragon curves using paper?

We will reveal the rules of dragon curves next week! Good luck!

Karen Ramsay-Smith : 18 Jan 2012 16:11 |  Tags: Challenge Festival Maths |  Comments (0) |  Close comments |  Trackbacks (1) |  Report a problem