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This question is based on the "Picture proof" challenges from Rankk.org...

IDEA: You want to hold up a painting using nails on a wall and string. The string is attached to the left and right sides of the painting and the nails are in between (let's say in a horizontal straight line for simplicity).

QUESTION: The object is to find ways of doing this to satisfy various requirements. For example, so that if any one nail is taken out, it will fall. Or so that if any two nails (but not one) are taken out it will fall. In particular, I want to find short solutions or even a method to construct all possible solutions. Basically, what exactly is the mathematical structure and what are the main results or whatever.

VISUALS: Here I use capitals for the inverses.

Incredible drawings

THEORY: First nomenclature. Let's say we have three nails: $a$, $b$ and $c$. If we pass over the first nail rightwards, we call that $a$, but if we go over it leftwards, we call that $a^{-1}$. Then if we conatenate or multiply them, it means we do one after another. So for example, $abc$ corresponds to the string going across over the top of all the three nails from left to right. In that case, all three nails would need to be removed before the painting fell. Something like $aba^{-1}$ means it's looped around $b$ but sort of hanging on $a$ as well. Only $b$ needs to be removed for it to fall.

Obviously $aa^{-1}$ cancels out. And $a^5$ means a bunch of loops. Removing a nail simply means removing all of its appearances in the formula. The painting will fall if the whole thing reduces to the identity. Eg) $bca^{-1}cac$ becomes $bc^3$ if the $a$ nail is taken out. Perhaps the most powerful tool I've found is that you can use a kind of conjugation as an "or" operator. It entangles the two parts. For example, $aba^{-1}b^{-1}$ will fall if either $a$ or $b$ are removed. The complement "and" operator is just multiplication. So $abcb^{-1}a^{-1}c^{-1}$ is $ab$ conjugated with $c$ and will fall if $c$ is removed or both $a$ and $b$.

As a starting point, who can find a short string for four nails that falls if any two are taken out?

PS. I can give more examples...

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    Interestingly, your OR operator applied to $a$ and $a$ gives the identity...2012-01-05
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    @RahulNarain Yeah, it's a special case which has equivalents in many areas of maths. Another "problem" is $a$ "and" $a^{-1}$. Both of them hold up a painting, but together, they collapse.2012-01-05
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    It might just be me, but I'm having a hard time visualizing what you are talking about. A picture with actual nails and string would be helpful.2012-01-05
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    @AustinMohr Fixed, I'm sure you're not the only one! abcABC is interesting because it needs exactly two nails taken out...2012-01-05
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    @user826788: That's interesting; can you name some of those equivalents?2012-01-05

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