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Problem of the Day #115: Picture Hanging July 12, 2011

Posted by Albert in : potd , trackback

There are $n$ evenly spaced nails on a wall, parallel to the floor. Saketh wants to hang a picture on the nails so that when all the nails are in place, the picture doesn’t fall, but when Sreenath removes any one nail, the picture does fall. The infinitely strong, infinitely thin, weightless wire of arbitrarily long length is already attached to the picture frame. Let $f(n)$ be the minimum number of times Saketh can do clockwise and counterclockwise wraps around the nails to set up this system (note: $f(1) = 1$). Find: $$\sum\limits_{i=1}^{32} f(i)$$


1. Rick - July 12, 2011

Clarification: Is “falling” defined as the picture frame no longer being parallel to the ground/vertical, if hanging a picture is defined as aligning the picture parallel to the ground/vertical?

2. Albert - July 12, 2011

Falling is when the string completely unravels around the nails so that the picture is no longer hanging.
The picture and the string will fall to the ground if set up properly, after one nail is removed.

3. Brian - July 12, 2011

What constitutes a “wrap”?

4. Albert - July 12, 2011

A “wrap” is at most one turn CW or CCW around a nail, and more than 0 turns around that nail.

5. Albert - July 12, 2011

Since so few people have gotten my problem, I’m going to give a little help:

The following gives you an idea of what my problem is referring to, although none of these images are solutions to any case of my problem:


The number of “wraps” is simply the number of letters in that diagram needed to represent a particular configuration.