Brain teasers

[twerppoet]

Seems selfish to solve puzzles, but not offer one, so:

An mild attack of insomnia led me to consider this common puzzle shown in figure 1. The idea is to draw it with one continuous line. Crossing lines is permitted, but retracing lines is not. If you've never seen it before, go ahead and solve it.

There are actually several solutions, I've worked out at least six; and that does not include mirror images and going backwards. In the process I noticed some interesting relationships, that on further thought made perfect sense. I've made kind of logic/geometry puzzle/quiz; so the sleepless hours were not completely waisted.


All solutions for figure 1 have one thing in common. What is it, and why?

Figure two has even more possible solutions, far too many for me to even bother working out. They also have one thing in common. What is it, and why?

If you've figured out the why answers to figure 1 and 2, then you should have no problem telling me why figure 3 has no solution.

View attachment 9162

Have fun.

Ok, I think it's been long enough. If you want to keep trying, stop reading here.

1: Every solution starts on one of the two bottom corners, and stops on the other. This is because those two points have an odd number of lines coming from them. The only way to create an odd number is to either stop the line there, or start it there.

2: Every solution ends at the starting point. Again, this is because you are drawing a single line. The figure only has points with an even number of lines coming from them. When you start a line it has an odd number of lines at that point. You must end the line there to make it even again.

3: This figure has four points where there are an odd number of lines meetings. The single line that you are drawing has only two end points, so you can only draw a figure with two odd junctions. You would have to lift your pen from the paper at least once to draw this figure.


Notes:

Another way of looking at this puzzle is as piece of string, wire, or straw that you can bend on a flat surface. When you look at it this way the two (and only two) end points become obvious, and your only two choices become obvious. Either the two ends meet and all points have an even number of lines leaving them, or they do not meet and there are two points with an odd number of lines leaving.

Once you realize that all solutions to figure 1 start and stop on the same two point it gets a lot easier to find those solutions.

When you realize that you can treat the x in the middle of the box as another point (instead of crossing lines), suddenly there are a 'lot' more solutions.

Unlike figure 1, you could start drawing figure 2 from the middle of a line. This means figure 2 has an infinite number of solutions. Even if you limit yourself to only starting where the lines meet, there are at least two solutions per point (probably more), adding up to a whole lot of solutions.


Here are some of the solutions for figure 1, just because.

 

[beavery]

He was bald.

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Got it in one!

 

[beavery]

The sides condense in?

I tried it with steam today. Put the lid on and the sides sucked in. Was a 3litre bottle...

 

[MrLuke]

My guess is it has to do with how much surface area of the hot water is exposed to the air. The more surface area the faster the transfer of heat from the water to the air, and the faster the air expands. The container should expand a bit even without any shaking, but the transfer of heat to air is probably not much faster than the loss of heat from the container, so it would lack drama.

On the other hand, if you let the air escape while shaking, the container would probably collapse shortly afterwards, as per the other guess.

Thought I'd point out that you're right on all that bar one thing. It doesn't effect the theory but it's worth remembering the fact that the air inside the container doesn't expand, it simply gains more energy, thus causing it to move faster and give the impression it's taking up more space.
An additional reason for the bottle expanding more with boiling water than hot water isn't just the reason that the air has more energy, it's also because the plastic bottle will have taken in some of this heat energy, which will make it more malleable and it will therefore expand further and faster.
I know I said I'd find that algorithm for you quite a while ago now and I will some time in the future but I've got exams to contend with at the moment.

Proud owner of a 32GB wifi iPad2 iOS5.


EDIT: @beaver, if you're wondering why this happens it is to do with the same principle mentioned above. When you put the steam in its taking up lots of space because all the particles have lots of energy. As soon as its in the bottle the steam starts to cool down. As it cools down it takes up less space. Technically this on its own wouldn't make the bottle collapse on itself, but because of the pressure difference between the outside of the bottle and the inside, the outside air forces it in on itself.

 

[Richard Brown]

Thought I'd point out that you're right on all that bar one thing. It doesn't effect the theory but it's worth remembering the fact that the air inside the container doesn't expand, it simply gains more energy, thus causing it to move faster and give the impression it's taking up more space.
An additional reason for the bottle expanding more with boiling water than hot water isn't just the reason that the air has more energy, it's also because the plastic bottle will have taken in some of this heat energy, which will make it more malleable and it will therefore expand further and faster.
I know I said I'd find that algorithm for you quite a while ago now and I will some time in the future but I've got exams to contend with at the moment.

Proud owner of a 32GB wifi iPad2 iOS5.

EDIT: @beaver, if you're wondering why this happens it is to do with the same principle mentioned above. When you put the steam in its taking up lots of space because all the particles have lots of energy. As soon as its in the bottle the steam starts to cool down. As it cools down it takes up less space. Technically this on its own wouldn't make the bottle collapse on itself, but because of the pressure difference between the outside of the bottle and the inside, the outside air forces it in on itself.

I was thinking along similar lines, as a lay person, in connection with the steam. I'm pretty certain I had read that steam can be used to cause a reverse reaction in a closed space, as against water expanding in a confined space, all connected to steam cooling and creating a partial vacuum?

Have you ever made jam and sealed the jar tops with a clear cellophane disc, secured with rubber bands, to each. When the jam, which is heated above boiling point, cools the cellophane contracts and a taught concave shape forms.

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[Richard Brown]

Here's another science experiment one, just to make Richard happy. This might require a minor financial investment.

Take a helium ballon on a car ride and hold or tie it so that it does not touch any part of the car. As the car accelerates, brakes, and corners the ballon will move.

How does it move in relation to how the car moves?

Is this what you expected it to do?

Why does it move this way?

I'll have to try this one out. I'm guessing that the balloon may not move too much as it is in an enclosed space and will move with it's supporting air.

I wonder how many of us have entertained the thought that if in a falling lift, would we save ourselves from excessive injury by jumping upwards just before the lift hits the bottom of the lift shaft!

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[twerppoet]

I'll have to try this one out. I'm guessing that the balloon may not move too much as it is in an enclosed space and will move with it's supporting air.

heh heh heh, you're going to like this one. It'll make you think.

I wonder how many of us have entertained the thought that if in a falling lift, would we save ourselves from excessive injury by jumping upwards just before the lift hits the bottom of the lift shaft! Greetings

Hey! It works for Bugs Bunny.

 

[55julie55]

You pesky wabbit!

 

[Bob Maxey]

My guess is it has to do with how much surface area of the hot water is exposed to the air. The more surface area the faster the transfer of heat from the water to the air, and the faster the air expands. The container should expand a bit even without any shaking, but the transfer of heat to air is probably not much faster than the loss of heat from the container, so it would lack drama.

On the other hand, if you let the air escape while shaking, the container would probably collapse shortly afterwards, as per the other guess.

When the water reaches equilibrium with the PTFE molecules in the plastic container, they form what are called "strong water polymer bonds." When these bonds are disturbed, they emit a vapor that will only react with water that has risen more than .05 percent of the outside temperature.


When you provide a sealed environment and you shake or otherwise disturb the long molecular bonds and chains, the back pressure causes the PTFE molecules to expand and occupy the spaces between the water molecules. The spaces begin to collapse due to entropy, thereby releasing one oxygen atom for every two PTFE molecules.


This release is converted to energy, the energy heats the water vapor in the container, leading to expansion. When this expanding molecules fail to reach equilibrium with both the water and PTFE molecules, it is said to be in Stasis and that means the water temperature will no longer increase, the PTFE molecules are reabsorbed, and the water cools.


As the water cools, the water vapor diminishes and pressure declines.

Then add ice cubes and a slice of lime and enjoy a refreshing drink.

 

[twerppoet]

Ok. I have to admit, I'd have never guessed (or figured out) that. It goes a bit beyond basic physics and my somewhat dated chemistry classes. Are you sure this is the primary cause? Heating a gas is normally more than enough to cause this kind of affect.

 

[Bob Maxey]

Ok. I have to admit, I'd have never guessed (or figured out) that. It goes a bit beyond basic physics and my somewhat dated chemistry classes. Are you sure this is the primary cause? Heating a gas is normally more than enough to cause this kind of affect.

LOL - Snicker - Laugh - Chortle

I made it all up. Every last word, just free rambling on and on.

 

[leelai]

LOL - Snicker - Laugh - Chortle

I made it all up. Every last word, just free rambling on and on.

Omg Bob you Larrikin, lol

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[twerppoet]

LOL - Snicker - Laugh - Chortle

I made it all up. Every last word, just free rambling on and on.

Hmmm. I should have known better.

 

[55julie55]

I'm lost??

 

[leelai]

Another Riddle

What makes a loud noise
While changing its jacket
Becomes larger but weighs less

The answer is in these letters
L. N M. U. C
V. E. V. O. S
O. P. G. R. P

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