Pure Maths: I have decided to better myself!

[0iD]

No it's 0iDology!

[Zak33]

hmmm....

OK, lets help here shall we, with the full answers:

Scientific Approach:

The scientist spends a long time, trying different combinations, working through to the theory that's it's impossible But he's gonna have to trust that no one ever miss proves him, because he's not tried every single combination. And that will take an eternity.

The Maths approach:
Develop an argument that will form a conclusion that is gonna be right forever. And dont even go near the chessboard to do it.

The corners of the chess board that were removed are both white
Therefore there are now 32 black and 30 white
Each dominoe covers two neighbouring squares which must always be different colours, ie black and white, there are no other options.
Therefore no matter how you lie them, even without doing any yourself, you KNOW that the first 30 dominoes MUST cover 30 white and 30 black squares

You'll end up with one dominoe, and two black squares. And you never ever get two black squares together.

It's impossible. Period. Solved and done with style.

[Zak33]

so why is this so key to me this week?

Because it shows that with the right brain work, you can PROVE something without doing it. And you can prove it 100% to the end of time. It wont be overturned.

Pythagorus Theorem.

In a right angled triangle, the square of the hypotenuse (the long side opposite the right angle) will equal the sum of the squared of other two sides. Always.

[Zak33]

now onto the beauty of Pi

I knew what Pi was from an early age. It's the number that, if you multiply it by the diameter of a circle, gives you the circumferance.

So, if your tyre measures 17inches from the tread on one side, right across the alloy to the tread on the other side, (that's the diameter of the whole tyre) you can mulitpliy it by Pi and youll have the circumferance (the length of how round it is)

This is how you work out putting 17" alloys plus high profile tyres on your Nova will ruin the speedometer reading when it used to have 13" wheels on it.

School boy stuff in theory.

except that Pi is a never ending number... it is an IRRATIONAL number.

3.141592654 in my head (memorised as a young boy) but it goes on longer than that.

Why is it beautiful to me now though?

Because I've never seen it expressed like this before

Pi = 4( 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + 1/17 - 1/19 + ......)

similar to this from wiki


that's beauty. The bottom line goes up in odd numbers forever and you simply change whether you are adding or subtracting the next one in the row)

[Barakka]

You can't talk about the beauty of numbers without Fibonacci

Knowing that a daisy will have 34, 55, or 89 petals as they're all Fibonacci numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89......

[JPreston]

My degree is in mathematics. I've read that Fermat book, but IMO the best and most brain-stretching book on maths/logic (with lots of computery science stuff too as it happens) is Hofstadter's "Godel, Escher, Bach" - I definitely, definitely recommend it.

[0iD]

hmmm....

OK, lets help here shall we, with the full answers:

Scientific Approach:

The scientist spends a long time, trying different combinations, working through to the theory that's it's impossible But he's gonna have to trust that no one ever miss proves him, because he's not tried every single combination. And that will take an eternity.

The Maths approach:
Develop an argument that will form a conclusion that is gonna be right forever. And dont even go near the chessboard to do it.

The corners of the chess board that were removed are both white
Therefore there are now 32 black and 30 white
Each dominoe covers two neighbouring squares which must always be different colours, ie black and white, there are no other options.
Therefore no matter how you lie them, even without doing any yourself, you KNOW that the first 30 dominoes MUST cover 30 white and 30 black squares

You'll end up with one dominoe, and two black squares. And you never ever get two black squares together.

It's impossible. Period. Solved and done with style.

Yes, but you could see it wasn't going to work just by looking at it, it's an engineer thing. No maths involved, purely common sense.

[Zak33]

but you should have been abe to explain it. In a way that completely rules out any debate. If you had done so, you'd have writtem a theorem, and that would have been perfect.

That's what maths seems to be about. I think it's intriguing in the extreme!

So this mutilated chessboard isn't maths in itself...but it explains the difference between science (your method of saying "its obviously not possible", without proof) and the mathematic approach.

[0iD]

but you should have been abe to explain it. In a way that completely rules out any debate. If you had done so, you'd have writtem a theorem, and that would have been perfect.

That's what maths seems to be about. I think it's intriguing in the extreme!

So this mutilated chessboard isn't maths in itself...but it explains the difference between science (your method of saying "its obviously not possible", without proof) and the mathematic approach.

Yes dear, sorry dear, won't happen again dear

[Zak33]

to further this thread, here is something I found equally intriguing.

"What is the least number of weights that can be used on a set of scales to weigh any whole number of kilograms from 1 to 40?"

Now I hasten to add that I am clearly a PC man and I struggled with this, as I worked immediately in binary!

[Zak33]

Yes dear, sorry dear, won't happen again dear

now go and wash your hands and be ready for lunch in 5 minutes

/tut

[Gerrard]

to further this thread, here is something I found equally intriguing.

"What is the least number of weights that can be used on a set of scales to weigh any whole number of kilograms from 1 to 40?"

Now I hasten to add that I am clearly a PC man and I struggled with this, as I worked immediately in binary!

Not sure if it matters, but are you allowed to put the weights on both sides? I.e. a 2kg one side and a 3kg the other side will allow you to measure 1kg.

[0iD]

Four, 1, 3, 9, 27.
No idea as to any set formulae, thats just thro a few scribbles & working on the principle Gerrard posted above. Sorry Zakky, don't put me back in the cellar... pleeeeeese.

[TheAnimus]

Imagine a smooth metal ring running around the earth's equator about the width of a sidewalk. Now imagine a metal cable wrapped around this first ring so tightly that nothing can squeeze between the cable and the ring. You may assume that the ring and cable are perfect circles even though they most likely would never be. Now this cable, which does not stretch, has three extra feet added to it and is made to magnetically float up and off of the ring so that extra three feet of slack is then equally distributed around the earth. This now gives you an equal distance between the ring and the cable anywhere you checked it around the planet. Now before you do the math, if you are going to do the math at all, what do you think the largest object is that will fit between the cable and the ring?

A) an electron (Less than 1x10-13 cm)
B) a virus (~100nm)
C) single skin cell (~20 microns)
D) a grain of sand (~200 microns)
E) a typical pearl (~9mm diameter)
F) a grape (~1 inch diameter)
G) a grapefruit (~5 inches diameter)
H) a pumpkin (~1.5 feet diameter)

This was the question our lecturuer opened the Modern Huristics course with, just to show how problem statement, is the most important in solving problems.

[Whiternoise]

I recall that question from a brain teaser site, i seem to remember the answer being a lot larger than you expect.. i'll go with H

[SiM]

A grapefruit?

I translate the question into this:
A circle. add 3 ft onto the circumference. find increase in radius.
notice that diameter increases by 3/pi (slightly less than 1)
so radius increases by slightly less than 1/2 a ft ~ 5 inches

Edit: why is there a long story to go with the question lol...