Brain Teaser No 1

2*2 is 4 which isn't the sum of two squares.

I think you are allowed to use zeros.

And I think you are allowed to accept that a = b from time to time.

Cheers

Don

Allowing zeroes means I can say that every sum of two squares is also a sum of three squares. Seems somewhat trivially profound...

But probably the start of reasoning towards the current problem.

Paul

Don,

I've filled a page of doodles (mostly right angled triangles) but still haven't a clue as to where to start on this one.

Dan

Re-cap,

....just to be sure we are all on the same wavelength.


if n = a^2 + b^2 ie if n is the sum of two squares,

we need to prove that 2n must also be the sum of two squares.


(simple) algerbra is all we are looking at....well (simple) as in 'once you've seen it'

of course, there could be many profound or geometric ways to help in the discovery of the appropriate (simple) algebra....

Cheers

Don

Don,
Still working on the proof, but the following is true I think:-

if the original equation is n = a^2 + b^2,

then 2n = (a+b)^2 + (a-b)^2

Steve D

steved has the proof.

I think more generally that,

(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2

the product of two sums of squares is itself the sum of two squares. Given that '2' itself is the sum of two squares, Don's special case is shown.

Paul

Paul,

steved has the proof.

...well, Steved certainly has the first and last lines of my algebra but he says he is still working on the proof.

My proof has two lines in between the two posted by Steved.

Cheers

Don

Squares

Steved, Paul, anyone else reading,

I suspect you are refraining from posting line-by-line proofs just to give each other and others a chance.....?

Or do you need a bit more time...

Or should I post the second line of my proof....

Or...

Cheers

Don

Golf

OK, here's one for all you golf fanatics (...the game, not the car....)

A golfer has six different clubs in his bag, only one of which is right for the shot to be played. The probability of him playing a good shot with the correct club is 0.6. (he's not the best of golfers....) The probability of him playing a good shot with the wrong club is 0.25.

If he plays a bad shot, what's the probability that he chose the wrong club.

Cheers

Don

Bad Shot

Just over 90% probability that he chose the wrong club?

Steve D

BAD SHOT

There are 2 ways of hitting a bad shot (with my golf, there are of course many more!!!):-

Right Club, bad shot. prob 1/6 x 0.4 = 0.0666667

Wrong Club, bad shot. prob 5/6 x 0.75 = 0.625


Therefore prob of the bad shot being caused by using a wrong club is


0.625/(0.0666667 + 0.625) = 90.4%

Steve D

Squares

Or should I post the second line of my proof....

n = a^2 + b^2

2n = (a^2 + b^2) + (a^2 + b^2)

= 3rd line....

Cheers

Don

Previously I presented the identity,

(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2

I'm not convinced it's necessary to prove the identity step by step but,

ac^2 + ad^2 + bc^2 + bd^2 = ac^2 + bd^2 + ad^2 + bc^2 + 2acbd - 2adbc

So it's shown that the product of two sums of squares is itself the sum of two squares. Now 2 is the sum of two squares, 1 and 1, so Don's original enjoyably obfuscated question is shown to be a special case of a generality.

Paul

Paul R,

Your generalised equation is a very elegant solution to my obfuscated little problem.

Since most of the other contributors to this thread are also ex rocket-scientists (or the equivalent) I accept that your post with the identity :-

(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2

didn't really need further explanation.

However, for my benefit, and for others who might occasionally glance at this thread, your subsequent line-by-line account was very helpful. And for the benefit of anyone who didn't observe it yet, (I'm probably talking to myself at this point) the "subtlety" of the proof lies with inserting :-

+2ab - 2ab (or in Paul's case +2abcd - 2abcd)

in the penultimate line. (see next post)

Of course, there will always be debate over just how many steps you need to show in mathematics. Paul is happy with the one-line identity; I am happy with the 4-line exposition below; Mrs D just doesn't see the point of it at all....

Cheers

Don

Squares

So, for the mathematically challenged, here is my 4-line proof:-

n = a^2 + b^2
2n = (a^2 + b^2) + (a^2 + b^2)
2n = (a^2 + 2ab + b^2) + (a^2 -2ab + b^2)
2n = (a + b)^2 + (a - b)^2

Cheers

Don

And for the even more mathematically challenged (since Paul already gave his solution) here is my explanation of Paul's ;generalised' teaser :-

(a^2 + b^2) (c^2 + d^2) = a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2 equation (1)

= (a^2c^2 + 2abcd + b^2d^2) + (a^2d^2 - 2abcd + b^2c^2) equation (2)

= (ac + bd)^2 + (ad - bc)^2 equation (3)

and as Paul says, by substituting 1 for c and also 1 for d you can reduce Paul's general equations to my more simple, enjoyably obfuscated, little teaser.....

Cheers

Don

Football Teaser....

...and the probability that England will ever win a penalty shoot-out is.......

Cheers

Don

Don

What about the answer to the golf question?

Steve D

What about the answer to the golf question?

Golfing crisis,

I have not seen a definitive answer to my golfing teaser.

My own calculation is set out below and matches Steve's.

I have also 'drawn' the problem to illustrate selection of 'wrong' or 'right' clubs and 'good' or 'bad' shots. Of course it yields the same answer because it's based on the same logic.

I accept Cliff P's complaint that you wouldn't choose a club at random, particularly for a short put. But this is abstract maths, not real golf...

Any challenge to my logic is most welcome....

Cheers

Don

Probability of picking the 'correct' club = 1/6
Probability of picking the 'wrong' club = 5/6

Probability of playing a bad shot with the 'correct' club = 1/6x(2/5)
Probability of playing a bad shot with a 'wrong' club = 5/6x(3/4)

Probability of playing a bad shot = (1/6x2/5) + (5/6x3/4)
Probability of this bad shot being due to selecting the wrong club
= (5/6x3/4)/ [(1/6x2/5) + (5/6x3/4)]
=75/83
=0.9
= Steved's answer

....and the picture version...

Cheers

Don

...oh dear,

I'll see if I can get the picture a bit bigger/clearer

Cheers

Don

hopefully....

Cheers

Don

Photos...

A real 'brain teaser' for me is how to get decent pictures up on this forum.

I don't have a digital camera, so I either have to scan (Epson GT7000) prints or transfer Kodak cd pictures.

The picture below was scanned at 50 dpi then transfered to a GIF format where it ocupied 49.7Kb of memory to just fit within the forum limit of 50Kb

If I try to make the picture bigger, I very quickly exceed the 50kb limit. Also the 50dpi resolution is the most coarse I can set.

How do others manage to post large, crystal clear well saturated images????

Cheers

Don

PS The 'Brain Teaser' is "where is this?"

Are you limited to the GIF format by the software you are using? If not then change to use JPEG (this is mostly a .JPG file on windows).

As a quick proof I saved your image to disk locally and indeed it is around 50Kb. I then saved it as a .JPG (just using ms paint) and it is approx 20Kb in this format.

I'm not that familiar with the GIF format, but I don't think it is compressed whereas the JPEG uses compression to store the image.

Compression...

Two-sheds,

Many thanks for your advice.

I have never bought a 'proper' piece of imaging software. I normally use a freebe that came with my scanner called Adobe Photo Delux. This allows cropping, rotation and some 'filtering' of the colours and a 'best image' button. I've never been impressed.

The first picture above was scanned from a 6" x 4" print at 50 dpi and the image size in the scanner reduced to get the file size down to about 250KB.

I then 'sent' it to the GIF file and saved it. This reduced the file size to just under 50KB.

I have now tried another bit of software called 'Imaging' (I don't seem to have MS Paint).

This time I have scanned at 200dpi and used the scanner to reduce image/file size to 1.33MB. Imaging has then reduced/compressed this file down to 45.1KB using JPEG (acording to the on-screen prompts). But it also refered to TIFF files = Tagged Image File Format ?)

Whether this produces a better image, or even any image I don't know....yet...

See below....

Cheers

Don