[Guest]

Hang a picture on a wall with N nails, such that, if any of the N nails failed, the picture would fall on the floor.

Remarks:

The picture is provided with a sufficiently long hanging thread, which together with the upper frame, forms a single loop.

Only the thread may touch the nails, and the picture is said to have fallen on the floor when the thread is not touching any nail.

Any nail could support the picture without the help of the other nails, if the picture is hung properly on it.

Edited December 14, 2010 by rookie1ja

[Guest]

Yes, it can be done for any N. Here's my drawing for N=2. Things get pretty messy as N increases.

Your picture would still certainly hang if the nail on the right were removed, provided there is enough distance to the floor (you have quite a long thread!)

[Guest]

Yes, it can be done for any N. Here's my drawing for N=2. Things get pretty messy as N increases.

This is correct! Well done!

No trickery involved, and certainly not impossible as some posts above suggested

Yes, it is possible for any number of nails N.

For the sake of completeness, if not described explicitely or drawn, please explain how the messy arrangement for higher N could arise.

[Guest]

from the original text:

..."if any of the N nails failed, the picture would fall on the floor."

..."Any nail could support the picture without the help of the other nails"

These two statements are self-contradictory, no?

Given the latter statement, the former can never be true...(i.e there is NO way to "hang a picture on a wall with N nails, such that, if any of the N nails failed, the picture would fall on the floor."

...unless I am missing/misinterpreting something...

I've got to go with Ring Ring and say that this is unsolveable for N <>1...

Smoo

If left out of the context, the 2 statements are contradictory. However, I mentioned that "Any nail could support the picture without the help of the other nails, if the picture were hung properly on it. "

The problem does not require to hang the picture properly on any nail, but the contrary, to hang it improperly so that it would fall if any nail failed.

It is not unsolvable, as Superprismatic showed

Try to find the flaw in your/RingRing judgement (hint: it has to do with false assumptions)

Edited December 14, 2010 by Kornrade

[superprismatic]

This is correct! Well done!

No trickery involved, and certainly not impossible as some posts above suggested

Yes, it is possible for any number of nails N.

For the sake of completeness, if not described explicitely or drawn, please explain how the messy arrangement for higher N could arise.

The simplest heuristic explanation up with which I can come at the moment is:

One nail will just have a single loop around it. The two strands which

come out of this loop must encircle each of the other nails. Furthermore,

they must go over and/or under each of the other nails in such a way that

the encircling mentioned in the previous sentence are locked in as long

as this original loop is held by its nail. The weaving must also be such

that the looping around each of the other nails lock in all of the others.

So, the number of loops grows as an exponential function of the number of nails.

I hope you enjoyed the grammatical construction of the first line!

[Guest]

is if one nail fails the picture would still be hanging by the other nail. Please explain if I am wrong.

is if one nail fails the picture would still be hanging by the other nail. Please explain if I am wrong.

Yes, it can be done for any N. Here's my drawing for N=2. Things get pretty messy as N increases.

[Guest]

is if one nail fails the picture would still be hanging by the other nail. Please explain if I am wrong.

If one nail simply disappeared, the thread would slip all around the other nail and the picture would fall. Try it at home: make the arrangement an see that the picture hangs; remove one nail and see that the picture falls.

Edited December 15, 2010 by Kornrade

[Guest]

If one nail simply disappeared, the thread would slip all around the other nail and the picture would fall. Try it at home: make the arrangement an see that the picture hangs; remove one nail and see that the picture falls.

Yes I see it now. Thanks.

[witzar]

The fundamental group of a plane with n holes is the free group of rank n. The rest is pure simple algebra.

[Guest]

If I'm following your description correctly, (I might have mis-understood) what you're describing is in diagram A.

[k-man]

Please tell me how to put an attached image into a spoiler like you did here.

I tried to do it for my last post but I couldn't figure it out. Thanks.

You upload the image, then click "Add to Post" link to the right of the uploaded file. It automatically adds the [attachment ...] tag to the post. Then simply move the [attachment...] tag inside the spoiler like this...

[spoiler='like this'][attachment=2742:frame1.PNG][/spoiler]