Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down

Question

A friend of mine told me that it's possible to hang a picture on the wall from a string using two nails in such a way that removing either of the two nails will make both the string and picture fall down. My friend also told me that I need to be acquainted with the concept of fundamental groups to understand the solution. The problem is that I'm not. Is there really no solution to this straightforward problem that doesn't require acquaintance with fundamental groups?

Answer 1

You can see a solution in Figure 1 of Picture-Hanging Puzzles, by Demaine et al. It doesn't require anything advanced to understand that specific solution. It's the generalization to more and more nails that seems to need some fancy math.

Here is the two-nail solution from Demaine et al. (don't mouse over it if you want to think about it first):

An excerpt of an explanation found in section 3.2:

We define $2n$ symbols: $x_1, x_1^{-1}, \dots, x_n, x_n^{-1}$.
Each $x_i$ represents wrapping the rope around [passing over top of?] the $i$th nail clockwise, and each $x_i^{-1}$ represents wrapping the rope around the $i$th nail counterclockwise. Now a weaving of the rope can be represented by a sequence of these symbols. For example, the solution to the two-nail picture-hanging puzzle shown [in the figure above] can be written $x_1x_2x_1^{-1}x_2^{-1}$.
...
In this representation, removing the $i$th nail corresponds to dropping all occurrences of $x_i$ and $x_i^{-1}$ in the sequence. Now we can see why [the figure] disentangles when we remove either nail. For example, removing the first nail leaves just $x_2x_2^{-1}$, i.e., turning clockwise around the second nail and then immediately undoing that by turning counterclockwise around the same nail. In general, $x_i$ and $x_i^{-1}$ cancel, so all occurrences of $x_ix_i^{-1}$ and $x_i^{-1}x_i$ can be dropped. (The free group specifies that these cancellations are all the cancellations that can be made.) Thus the original weaving $x_1x_2x_1^{-1}x_2^{-1}$ is nontrivially linked with the nails because nothing simplifies; but if we remove either nail, everything cancels and we are left with the empty sequence, which represents the trivial weaving that is not linked with the nails (i.e., the picture falls).

Answer 2

How about puting two nails on the wall so that the picture is resting on top of the nails and there is no nail in the middle? Won't that do it?

Answer 3

The solution can be understood without directly referring to the fundamental group (but knowing about fundamental groups makes it much easier to generalize to more nails).

Just consider wrapping the string once around one nail, once around the other and then the same again, but wrapping it around in the other direction (but still with the nails in the same order).

Answer 4

Put one nail in the wall, and another one in the picture frame. Wrap the string around them. If you remove any of the nails, the picture will fall down :)

Answer 5

Assuming the picture is a square, rotate it 45 degrees so it now has the shape of a "diamond" (rhombus). Now place a nail just under and a bit to the right of the left corner, and another nail just under and a bit to the left of the right corner.

Answer 6

Nail the picture on the center of left and right sides. The 'nail hold' of the frame have an opening perpendicular to the side, whose width equals the diameter of the nail. When one of nails are removed, the frame rotates till the nail reaches the opening. Once it have reached the opening, the nail will slipde through it and the picture will fall down.

Answer 7

Imagine having two nails in 3d, positioned perpendicular to the face of the wall (i.e., nails are horizontal), but with the two nails in perpendicular directions to each other and one nail above the other nail. It should be much easier to come up with a solution for how to wrap the string in this visualization so that the string falls when you remove either nail.. Then, once you have this solution, all you have to do is rotate one nail so that they both face the wall. This gives exactly the picture presented by Barry Cipra.