Brain Teaser No 3

quote:

Originally posted by AlexanderVH:
BTW, Dun Eideann, isn't that Edinburgh? It's on the bottle I'm tasting. I've got 10 bottles of whisky to taste and it's hard work.

Yes Dun Eideann is the Gaelic name for Edinburgh (and where Dunedin in NZ got it's name). Tread gently with the good stuff, it can affect your puzzling-skills!

Ian

I appreciate good whisky but can't seem to get round to tasting it all. The box is still standing there, waiting.

Alright, pens down in 15 minutes!

That's it then. I couldn't come up with anything pretty in those 15 minutes either. did some attempt to convert the edges to morse and single out the vowels.

quote:

I suspect answers in the form of a negative are stronger: dot dot is the odd one out because it's the only one that is not dot dot dot. (Hi there, Don

Glad to see you all enjoying yourselves...unfortunately I have been incredibly busy and will continue to be busy for some time. Don't take my lack of participation as a lack of interest. When time permits, I do take a quick look.

Cheers

Don

This one had me in stitches:

Grandmother uses her old sewing machine to stitch a new zipper in your trousers. The sewing machine uses the standard stitch(it only has one stitch) and goes down 15 cm, then 2 cm sideways, then 15 cm up. With one stitch per 5mm, how many knots are there in the sewing threads?

And yes I just made it up so let's hope I didn't overlook anything.

15cm + 15cm +2cm =32cm / 0.5 = 64 knots

64 + 1 at the beginning ? Not sure I know enough about sewing to answer the question!

Ian, when we are done, sewing machines won't have any secrets for us anymore
Meanwhile I found that there are two candidates for our old stitching style and they yield different results. See the cute animations at How Stuff Works.

There is the chainstitch which can unravel completely if you pull on it, indicating it has no knots at all. That was the machine I had in mind. The lengths were there just to point you in the wrong direction

The lockstitch on the contrary cannot unravel and apparently is more common than the first one. We live and learn. The lower thread actually does pass through the loops the upper thread makes and for each stitch the only way to free the lock thread is to cut the top thread or to pull on the lock thread.

So the answer is either 64-65 or 0, depending on the technique.
The definition of what is a knot could also benefit from a little scrutiny.
I considered something like this. Assume that we're holding the upper and lower threads by their ends.
The knot count is then "the minimal amount of times you have to let go of an end and pick it up again in order to untangle it from itself as well as from the other thread."
You can also use "the minimal amount of times a thread has to cross another thread (cut-and-mend operation) to be freed."

Which brings us to..
There was once a poor but brilliant mathematician who boasted he could create a sheet of cloth
from a single thread without making any cut or knot in it...
What did he invent? oh well, maybe just one or two knots.

quote:

Originally posted by AlexanderVH:

Which brings us to..
There was once a poor but brilliant mathematician who boasted he could create a sheet of cloth
from a single thread without making any cut or knot in it...
What did he invent? oh well, maybe just one or two knots.

Does the answer involve the phrase "But he has nothing on"! And perhaps the expression "The Emperors New Clothes"?

Beano

Almost, Beano. He has knitting on, no knotting. With knitting you can make a sheet from a thread, and since you can unravel the whole sheet again by pulling on an end, one concludes there are no knots.

The knot at the start and at the end are there to keep the thing together. It may depend on the style of knitting again. Alas, it is something I know knothing of.
It's interesting - to some - that knitting ought to have been invented by a mathematician, but it was not.

So we have learneth that a knit hath kno knots..

Here's a mathematical cutie I heard for the first time yesterday.

What is the value of i^i , I mean i the sqrt(-1) raised to the power of i.

Ian

This is a good one. The answer comes from Euler's formula:

e^(i*pi) + 1 = 0

Subtract 1 from each side, substitute pi for (pi - 4*n*pi), where n is an arbitary integer, square root both sides, and finally raise each side to the power of i.....this yeilds:

e^(i*i*(pi/2 - 2*n*pi) = i^i

So i^i has an infinite number of values of the form:

e^(-pi/2 + 2*pi*n)

All good fun! One may first think that the answer is just i^i = e^(-pi/2). But this does not account for the 'cyclic' nature of the Argand diagram where any solution plus 2*n*pi is also a solution.

Please forgive any mistakes....it's hard writing out maths like this!

Nice work Ben, that's probably more complete than what the originator of the question had in mind

I would have done the following.
- First remember what complex numbers are again. The rest is easy.
a+ib=r*e^i@=r*cos@+r*i*sin@
- From this follows that i=cos(PI/2)+isin(PI/2)=e^(iPI/2)
Concluding
i^i=e^(i*i*PI/2)= e^(-PI/2) = approx 0.2

Subsituting @ by @+2nPI for @=2PI in the last step generates the other results.

Well done Ben, & Alex, I didn't really expect anyone to worry about the multiple branches but sometimes one can't suppress ones education !

I think it is really cool that i^i is a real number ( or one of several ).

Ian

The Cutlery Affair(also from an old de Bono book).

If you already know the answer to this one, please restrain yourself Requirements: 4 glasses. Also useful: 4 knives. The task is to place 3 of the glasses in an equilateral triangle and suspend the fourth glass right in the middle of the triangle, at 'glass height' let's say.

It's strongly recommended to make use of knives in order to keep the 4th glass up in the air, but not to make things too easy the distance between the glasses (their top edges, not their centres) has been made 25% larger than the length of the knives. There is no intention that any kind of wordplay would be needed to solve the problem.

Maybe everyone knows it then. Hint: you need three knives. The fourth knive is needed for the followup question.

Haven't had the courage to try this (SWMBO is at home !)

How about 3 glasses, open ends up in the equilateral triangle, one knife in each glass forming a tripod (maybe part way up the glass), 4th glass inverted on top of the knives pinning them down and outwards.

Ian

It might work. But I know a solution that is much more esthetically pleasing.