Two simple puzzles.

Hobbes

Puzzle #1.

You have 5 links of a chain...

O O O O O

It costs $1 to open a link, $1 to close a link. What is the least amount of cash it would cost to join all links to make a circle (bracelet).

V = Open
O = Closed
- = Joined.

So ....

O O O O O
Should turn into.
~O-O-O-O-O~

Puzzle #2

Draw a 3x3 Dot matrix like so..

O O O
O O O
O O O

Now draw no more then 4 straight lines that will cross through all 9 dots.

to make it easier... give each dot a number.

1 2 3
4 5 6
7 8 9

So if you draw from top left to bottom right you would say 1-5-9

Oops... Once you solve 4 lines, try to do it with 3.

When you solve the 3 lines, try to do it with 1.


[This message has been edited by Hobbes (edited 08-09-2000).]

MiCr0

the second one is easy
1-4-7-a
a-2-6-b
b-9-8-7
7-5-3

where
a
1 2 3
4 5 6
7 8 9 b

MiCr0

the first one can be done in $8 any way
but i'm sure its meant ot less

UNIFLU

Least amount $2..

open 2nd and 4th allowing you to join all together....

o v o v o

o-o-o-o-o

UNIFLU

damnit i cant add $2 and $2 is $4..

UNIFLU

I presume the lines have to be touching end to end like the idea of not lifting the pen. If so how can you do it with 3 lines ?

Hobbes

That is correct, the pen can't leave the page. Sorry I forgot to mention that.

UNIFLU

I seriously dont think it can be done with three continual lines. Youd have to bend the lines or move the dots as you need an average of three points per line and that cant be acheived as far as i can see.

Hobbes

Ooooh it's possible alright It is also possible with 1 line.

Hobbes

Originally posted by UNIFLU:
damnit i cant add $2 and $2 is $4..

Also you didn't link the outside chain together to make a circle.

Hobbes

V.Good Mirco, now do it with 3 lines.

jahras

You have a jug that holds five gallons, and a jug that holds three gallons. You have no other containers, and there are no markings on the jugs. You need to obtain exactly seven gallons of water from a faucet. How can you do it?

Hobbes

You watch Die Hard III for the answer?

Quit stalling the lot of you

Hobbes

Ok looks like no one is going to get the 3 one.

So here is the explanation.

The line goes...

1-2-3-a
a-6-5-4-b
b-7-8-9

This is perfectly possible if you do not draw the lines through the center of the dots.

pat kenny

Fill the 5gallon jug right to the top from that fill the 3 gallon jug right to the top then empty the 3 gallons away you are now left with 2gallons in the 5gallon jug pour them into the 3 gallon jug now fill the 5gallon jug right to the top now there.
update:5gallons in the 5 gallon jug
2gallons in the 3gallon jug
fill the remaining letre of the 3gallon jug from the 5gallon jug leaveing exactly 4gallons in the 5gallon jug.

Take it *****.

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O man it's total grid-lock.

TonY Stark$

sorry im not sofisticated for any of this or else i just couldnt be arsed

Hobbes

So no one can solve the puzzle using only 1 line?

MiCr0

hobbes that 3 line soln doesn't work if u just use single dots. :P

UNIFLU

Hobbes,

Its either gotta be a damn bendy line or youre gonna have to shift the dots

Hobbes

No no no,

I didn't say what size the dots were, lets sau make them the size of a penny and 2cm apart, it will work fine if you don't draw through the center of the dots.

The whole point of the puzzle is to show that people define boundries to thier thinking before they even start to solve the puzzle.

Take for example the 4 lines. Most people will not draw past the cube layout. In the 3 dots most people will always draw through the center of the dots regardless of size.

Now work out the one line.

Shiminay

Ah - so the dot's can be huge and can overlap one another?

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All the best,

Dav
@B^)
My page of stuff

Hobbes

Originally posted by Kharn:
Ah - so the dot's can be huge and can overlap one another?

No. They can be any size, but they have to be evenly spaced

Cerberus

This might be an answer to the case using only one line.
This mightn't be the only answer but it is an answer...hopefully. There probably is a far easier one. Ok here I go....

What people see as a line is only relative to the their position when viewing the line.
It is dependent on what plane you are on. i.e.
A sheet of paper seen from the side appears to be a line.
So if you take the 3 dots as being three marbles evenly space out on level ground and then place a sheet of paper down on top of the spheres(marbles). Put your head on the ground and look in from all four sides and what you see on each side is a line touching 3 spheres at one point(dot) on each sphere.
Take a look in from the corner and you will see the line is also touching the middle sphere on one point. Putting all this together from this view point and one can see 'plainly' that there is a line touching all 9 spheres.
Now replace the marbles with dots or pennies or whatever.QED
Christ - what depths of my brain did I pull that outta??

Hobbes

Excellently done and completly wrong

Although you were thinking the right way. By wrapping the paper around a cylinder it is possible to draw a single straight line that spirals around to pass through all dots.

Cerberus

Cool. I should have thought of that.
There should be more of them puzzles put up.
After posting my bit I was later thinking that it was wrong, totally and absolutely plain wrong. Just as well I'm not a maths student or I'd feel a like bit of a turkey.

Hobbes

Originally posted by UNIFLU:
hobbes,

im afraid i have to complain about your three lines as if you pass through ant two points (not dots as you ended using) you cannot possibly return to the next row. i kept thinking about points not dots so couldnt get the answer.

I said to pass through the dots. Not the points. Don't feel upset though, most people think points or the center.

Here is the answer I have made.

Cerberus

To understand this when the circles are only dots think of this way.
A dot is infinitely small and a line is infinitely thin so when you scale the line and cirlce down to these infinite proportions Hobbes diagram will still work.
You can also do what Hobbes did without passing the lines through the cirlces. It merely has to touch the sides of the circle so that the lines are at a tangent to all the circles. I will post a picture when I get a chance.

UNIFLU

Hobbes,

No worries, thanks though it kept me busy in work for a while. The one line solution is damn good though.

WhiteWashMan

well, you could use a big paint brush that covers everything at one sweep, or you could look at the coins from the side, ie in 3d an then all you have to do is draw though the 3 you see, but you actually cover the whole 9

Hobbes

Originally posted by WhiteWashMan:
well, you could use a big paint brush that covers everything at one sweep, or you could look at the coins from the side, ie in 3d an then all you have to do is draw though the 3 you see, but you actually cover the whole 9

lol I like your answer, keeps in the same theme.

UNIFLU

hobbes,

im afraid i have to complain about your three lines as if you pass through ant two points (not dots as you ended using) you cannot possibly return to the next row. i kept thinking about points not dots so couldnt get the answer.

C B

Originally posted by Hobbes:
Excellently done and completly wrong

Although you were thinking the right way. By wrapping the paper around a cylinder it is possible to draw a single straight line that spirals around to pass through all dots.

can a spiral be considered straight?

Hobbes

Originally posted by C B:
can a spiral be considered straight?

Everything is relative to your frame of reference.

While a spiral doesn't look straight when you straighten out the page, it is straight when you have it wrapped around the cone.

C B

Originally posted by Hobbes:
Everything is relative to your frame of reference.

While a spiral doesn't look straight when you straighten out the page, it is straight when you have it wrapped around the cone.

i would have thought the exact opposite. the line is straight when the paper is unfurled. but when wrapped around the cylinder or cone (which are entirely different objects) it loses its straighness.

Myaybe i'm being a complete ass and can't see what right in front of my face.

Hobbes

The puzzle is taken from a book which I'm currently reading (along with others). It's called "Psychology of Intelligence Analysis" by Richards J. Heuer, Jr. (who works for the CIA).

It explains it in more detail, but basically if I draw a straight line from A-B (from top to bottom) while the cyclinder is rotating it creates a spiral.

The puzzle in its self is not to solve the answer but to prove that people set limitations on thier answers based on thier biased outlook on life.

So the ball bearing answer in a way is correct, if I drew 3 lines and asked someone else to join the lines together that answer would be techincally correct based on the orginal rules given.

These sort of thinking can be put to real world applications in development. Too often a developer will say "That can't be done" or suggest a textbook method to solve a problem when trying to think outside the rules works out better.

C B

yes,
i agree entirely with you. too often people fail to find solutions by not looking at a problem in a more holistic fashion.

this however does not solve my difficulty with your answer the spiral is not a straight line maybe you need to fax me the page from the book.
wait soory i'm an idiot i get it.

[This message has been edited by C B (edited 21-09-2000).]

Hendrix

No Spiral is not straight, he/she is gay!!

Q_Ball

<font face="Verdana, Arial" size="2">Originally posted by Hobbes:
Ok looks like no one is going to get the 3 one.

So here is the explanation.

The line goes...

1-2-3-a
a-6-5-4-b
b-7-8-9

This is perfectly possible if you do not draw the lines through the center of the dots.
</font>

<edit> dont mind me i'm a ranting fool. i get it now </edit>

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It's a 106 miles to Chicago. We've got a full tank of gas, half a pack of cigarettes, it's dark and we're wearing sunglasses.

HIT IT!

[This message has been edited by Q_Ball (edited 03-05-2001).]