I finally worked it out

This puzzle has bugged me for years. Take a right angled triangle and split it up like this, call it triangle1

Now rearrange the pieces like this to give triangle2

The overall triangle is still the same size 13 by 5, none of the pieces has changed, so where does the hole come from?

Not convinced? Here’s an animation.

Draw it out yourself and cut it up with scissors. It works. It is true, or so it seems. Of course it's a subtle trick. It all looks convincing to the eye and, given the accuracy of drawing on graph paper and cutting with scissors, it works on paper.

The solution? The easy way to see it is to superimpose triangle1 on top of triangle2.

Hey! there's a bit left stuck out along the hypotenuse - that bit is the trick. Here's a blowup:

In truth, triangle1 is a smidgeon smaller than a real 13 by 8 triangle, triangle2 a gnat's hair bigger. That slender difference, surprisingly, adds up to 1 and that's the size of the hole.

Just as a last observation, look at the whole (integer) numbers needed to make these shapes.

The main triangle is 13 by 5
red is 8 by 3
blue is 5 by 2
yellow and green have sides of 1, 2, 3 and 5
the hole is 1 by 1 but really zero

Er...
0,1,1,2,3,5,8,13

Where have I seen that before? Of course, now that's interesting!
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svg code for images adapted from here. See also Missing square puzzle. Grow into these trousers... >>

Geometry in Nature

Many things in the natural world have a simple underlying mathematics, and we often miss it unless our eyes are open; but the symmetry is always there.

We are all familiar with the interlocking hexagons of a honeycomb, made from equilateral triangles.











But there is another mathematical trick which Nature uses in some surprising ways. The shape of a snail's shell, the scales on a pine cone, the whorls in a flower, the breeding of rabbits (it's true, see below). Much more, and it all follows a simple mathematical sequence -
The Fibonacci Series.

It goes like this:

You start with nothing, zero
0
along comes a 'first', the number one
0, 1
add them together 0+1=1
0, 1, 1
then add 1+1 to get 2
0, 1, 1, 2
keep adding the last two numbers, so 1+2=3
0, 1, 1, 2, 3
then 2+3=5
0, 1, 1, 2, 3, 5
and so on...
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

If you divide each number in the sequence by the previous one* you get increasingly more accurate approximations to one particular number...

1÷1=1
2÷1=2
3÷2=1.5
5÷3=1.6667
8÷5=1.6
13÷8=1.6250
21÷13=1.6154
34÷21=1.6190
55÷34=1.6176
89÷55=1.6182
144÷89=1.6180...

That number is the Golden Ratio, that pleasing blend of height and width beloved of artists and architects for millennia. But Nature has been using it for far longer:

Take the Golden Ratio and make a triangle where, if the base equals 1 then the long sides are 1.6180. A Golden Triangle, Now let's play by putting them together...










And again you can join the dots to find a snail shell:

Now watch this short animated video by Cristóbal Vila which explains things in a beautiful way.

Nature by Numbers [3:44]


Oh! I promised rabbits, here you go...

Fibonacci with RABBITS!!!! [1:57]

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*Avoiding division by zero (1÷0 is not infinity, it is just not a number, it's undefined and so it's the worst approximation you can possibly get) Grow into these trousers... >>

E = mc2(1-v2/c2)-½

I've been following a vast series of maths lectures over the last 6 months or so from the physicist Leonard Susskind. This is a guy who can argue the toss with Stephen Hawking and then go teach a class of continuing education students. It's the recordings of his CE classes that I'm watching.

Susskind has a such a likable personality and easy style he can often lull you into a false sense of security, and then he'll snap you out of it with a jerk. I wish I'd had a teacher like him 25 years ago when I still could do maths properly (practice, Holroyd, practice - yes I know).

Fast forward this lecture to about 1h 15min and watch him derive Einstein's famous E=mc² with such consummate skill it's like he's stating the bleedin' obvious. His play with 'c' the speed of light, is almost comedy and his correction by (1-v²/c²)^-½ is the bit non-physicists always forget (but that's the most important bit 'coz it tells you it's impossible to accelerate a mass to the speed of light*).

Lecture 6 | Modern Physics: Special Relativity (Stanford)

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If you are daft like me and want to follow the whole thing, start with his course on classical mechanics:
Lecture 1 | Modern Physics: Classical Mechanics (Stanford)
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* I can't think how to html 'one divided by square root' except to raise it to minus the half power Grow into these trousers... >>

A mathematical joke

OK this is not funny, just an absurdity caused by incorrect thinking.

Take the number 1 and double it, you get 2. Double it again, that's 4. Double again, 8, again, 16 and so on...

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 ...

I seem to remember a story in Chinese history where an Emperor was duped by an Old Man and a chessboard. For some payment the Emperor agreed to place 1 grain of rice on the first square, 2 on the second, 4 on the third, 8 on the fourth and so on. Then the 64th square alone needs 9,223,372,036,854,775,808 grains of rice. Add up all the rice and it's twice that plus 1 (or is it -1) anyways it's about 18,446,744,073,709,551,616 grains. An impressive harvest.

However, had I been the Emperor I would have said "No. It's not enough payment. Let us make the chessboard infinitely large. How much do I owe you now Old Man?".

Let's work it out. We don't know what the final sum of all the rice grains will be so lets call it P, for Payment.

So to find P we have to add up the grains on every square, so

P = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 etc, etc to infinity.

How the hell do you do that? Think this way...

Double everything, ie multiply P by 2 and all the numbers by 2, then:

2P = 2 + 4 + 8 + 16 + 32 + 64 + 128 etc, etc to infinity.

Notice that each number in 2P has a counterpart in P, except for one (assuming the sequences continue to infinity) and that solitary digit is the number 1. It can't be in 2 times anything or it would be two.

Let's work out what the payment is. Subtract 2P from P, what do you get? What else can you get but 1?

P - 2P = 1

Therefore P = -1

"Old Man, you owe me a grain of rice"

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So where does the math go wrong? Grow into these trousers... >>

Thunder and lightning - part II

Glance back to yesterday's post to get the scenario.

So Mr Smokie had counted 7 seconds from seeing the lightning to hearing the thunder, and said it was 7 miles away.

Something told me this was wrong, a gut feeling if you like, and then I remembered somewhere I know well. It has an echo! The only place the echo could come from is a row of terraced houses a couple of hundred yards away (there is nothing else, just open fields), but the echo is loud and distinct. Clap hands and about a second later it's repeated back to you. Remembering that the sound has to travel maybe two hundred yards, bounce, and return back; then a crude estimate of the speed of sound might be 400 yards in a second. This slowly percolated into my brain.

But then, due to alcohol induced delirium joining in with the general early-evening banter around the bar, it took ages for these thoughts and numbers to congeal...

1760 yards to the mile (call it 1800), divide by 400 yards per second gives about 4½ seconds per mile. Mr Smokie was way wrong - 400 yards per second times 7 seconds is 2800 yards. Just over a mile and a half. That was eventually my guesstimate. Of course by this time Mr Smokie had buggered off and my taxi was due. So I went home and then just had to look it up.

The speed of sound in air? Can vary from about 330 to 340 meters per second and, as RBH pointed out, it depends on pressure, temperature and (I'd never have thought this) humidity; which I guess changes the overall density of the air by adding water vapour.

For simplicity lets say air pressure is normal, it's a bit above 0°C and not damp. The speed of sound works out to be 333.33 meters per second. You may guess where I'm going, that's 3 seconds per 1000 meters. Or 3 seconds per kilometer.

Mr Smokie counted 7 seconds, multiply by 333.33 meters per second gives you 2333.33 meters or 2.3 kilometers, give or take a few yards.

That's 1.45 miles.

Wahhh! Across the valley that's about where I liv.. Oh, other direction.
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And RBH was right:
5s x 333m/s = 1666m = 1.035mi
Grow into these trousers... >>

Thunderstorms - how near is the lightning?

There was a thunderstorm this tea-time. I was safely sat at the bar but one of the crew had gone outside for a fag (I just gnawed on my 'chomper') when there was a loud boom of thunder.

Mr Smokie came back in. "That's 7 miles away is that" says he, "count t'seconds after t'lightnin' an' that's how many miles". Or words to that effect. I think there was richer linguistic use of the term 'fuck' (find out more here).

Now; I remember counting from the lightning when I was a kid but, sat in the bar tonight, I couldn't remember just what relationship was. Till I worked it out.

Any answers? Grow into these trousers... >>

The solution to yesterday's teaser

Skip back to yesterday if you missed it.

How old are Jane's kids? OK...
Jane tells John "the product of their ages is 36"

How many different ways can you multiply three whole numbers together to get 36? Let's try:

1 x 1 x 36 = 36
1 x 2 x 18 = 36
1 x 3 x 12 = 36
1 x 4 x 9 = 36
1 x 6 x 6 = 36
2 x 2 x 9 = 36
2 x 3 x 6 = 36
3 x 3 x 4 = 36

Jane says to John that "the sum of their ages is the same as your house number."

We don't know where John lives but let's add up the possibilities anyway:

1 + 1 + 36 = 38
1 + 2 + 18 = 21
1 + 3 + 12 = 16
1 + 4 + 9 = 14
1 + 6 + 6 = 13
2 + 2 + 9 = 13
2 + 3 + 6 = 11
3 + 3 + 4 = 10

I hope John would know his own house number, but he says he still can't work out the kids' ages. How can this be?

Well, if John lived at number 36, 21, 16, 14, 11 or 10 he would have worked it out by now. There's only one sum for each where the ages multiply up to 36. ie if John lived at No. 14 he would know Jane's children were aged 1, 4 and 9.

So John must live at number 13.

We now have two possibilities: 1, 6 and 6 or 2, 2, and 9.

Jane tells him that "The oldest one has red hair." So the children can't be aged 1, 6 and 6 or there would be two oldest.

So Jane's children are aged 2, 2, and 9.

Hope I didn't cause too many headaches :)

The problem for next week is to come up with a solution to the world's financial crisis so I can take early retirement this year with no loss of income. Should be easy after this warm up.
Grow into these trousers... >>

Another brain teaser

I enjoyed the maths problem last week, it's good to keep practising these skills. Here's another little teaser to confuse and amuse.

Two friends meet and part of their conversation goes like this:

John: "I can't remember how old your three children are Jane."
Jane: "Well, the product of their ages is 36."

John: "I still can't think what their ages are."
Jane: "The sum of their ages is the same as your house number."

John: "I'm still not sure"
Jane: "The oldest one has red hair."

John: "Ah, of course. Now I know."

How old are Jane's children?

Answer tomorrow. Grow into these trousers... >>

The solution to yesterday's math problem

Just to put yorksnbeans out of her misery...

Have a look back at the question if you missed it.

For starters let's call the number of gold coins they find p for purse, and let the wealth of each of the three merchants be x, y and z gold coins.

So from the problem we're told

x + p = 2(y + z)
y + p = 3(x + z)
z + p = 5(x + y)

and we have to find values which satisfy all three of those equations.

First write those sums in terms of the purse

p = 2y + 2z - x         --- (1)
p = 3x + 3z - y         --- (2)
p = 5x + 5y - z         --- (3)

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Let's get rid of the purse for now by taking (3) minus (2)

p - p = 5x - 3x + 5y + y - z - 3z

add up the like terms

2x + 6y - 4z = 0

and divide by 2 for simplicity

x + 3y - 2z = 0         --- (A)

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Do the same again with (3) minus (1)

p - p = 5x + x + 5y - 2y - z - 2z

so:

6x + 3y - 3z = 0

This time divide by 3 to make it simpler

2x + y - z = 0         --- (B)

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Now, 2 times (B) minus (A) gives

4x + 2y - 2z - x - 3y + 2z = 0

so

3x - y = 0

or

y = 3x         --- (C)

and 3 times (B) minus (A) gives

6x + 3y - 3z -x -3y + 2z = 0

so

5x - z = 0

or

z = 5x         --- (D)

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Gold coins come in whole numbers (usually), the smallest whole number is 1.

Looking at (C) and (D) tells us that x has the least gold, so let x = 1. Then (C) and (D) give values for y and z, and plugging those values into (1), (2) or (3) gives us the purse. Result!

x = 1
y = 3
z = 5
p = 15

Of course x could be any whole number, there is no unique solution, ie:

Great fun! Thanks to Mr P at the bar for setting it. I'll look out for more little posers like this, it keeps the grey cells active.
Grow into these trousers... >>

A math problem

The things I get asked in the pub!

"Oh - Andy, can you help me with this little problem? I've tried but I got stuck. It's from the Indian mathematician Mahavira, from about 900AD."

Seriously, I was asked this whilst sat at the bar. I like this place!

Three traveling merchants find a purse full of gold coins on the road.

The first merchant says “If I had the purse, I would have twice as much money as you two put together.”

The second says “If I had it, I would have three times as much as both of you put together.”

The third merchant says “If I had it, I would have five times as much as you both put together.

How much did each merchant have and how much was in the purse?

It took me a side of A4 to get the answer but I'm sure it can be done more succinctly... anyone?

Please show your workings.
I'll post my solution tomorrow. Grow into these trousers... >>

What do mathematicians do?

Judging by this they make impressive videos presentations. Credit to: Étienne Ghys of the École Normale Supérieure in Lyon, France. See Science News for a good article on this video.

Dimensions
Great graphics and big maths fit together so well.

Starting with projections of the spherical Earth onto a flat plane, that's 3 dimensions displayed in 2, the video takes you through the steps to visualising 4 dimensional objects by projection into 3.

Later, in part 5, they move on to complex numbers, which are just like normal numbers but with an imaginary bit, a multiple of the square root of -1, this adds extra dimensionality to all numbers (the imaginary part is at 'right angles' to the normal number line, this gives the complex plane).

On to fractals where the easiest way to think of them is in the complex plane.

Through topology, described by intersecting complex planes. Doughnuts seem popular here.

Finally they consider the work of Bernhard Riemann some of whose geometries paved the way for the young upstart, Albert Einstein.

Happy viewing.

via Jim Downey at Unscrewing the Inscrutable.
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I was just recovering from learning overload when I found they are planning 'Dimension II' - whoah!!!

Consider a spherical cow... Grow into these trousers... >>

Robot makes Rubik's Cube look easy

A robot made from a Lego kit has been built to solve a Rubik's Cube.
Brilliant! Full story at the Telegraph link below.

Video from brightcove.com via telegraph.co.uk via Skepchick.

Game over! Grow into these trousers... >>

The origin of the number 1

A light hearted Terry Jones documentary about counting.
The Story of 1 (BBC - 2005)


via Online Documentaries 4 U
NOTE: this looks like a great blog for scoring your documentary fix. Grow into these trousers... >>