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Three nails are placed on a wooden board. You are given an elastic band. Can you stretch it around the nails so that it relaxes when any of the three nails is removed? The band is relaxed when it does not surround any nail, and it is stretched otherwise.

(Non)example:

Here’s a configuration where the band is stretched when all three nails are present. However, when a nail is removed, it remains stretched, so this doesn’t answer the question.

A band is stretched around three nails forming a triangle

Warmup:

When there are only two nails, you can stretch the band around them as shown below. After releasing the band in this position, it remains stretched, but if you remove any of the two nails, it relaxes.

band stretched around two nails that relaxes when one of the nails is removed

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  • $\begingroup$ I must be picturing this wrong because how does the rubber band cease to surround the other two nails when one is removed? By "surround a nail" do you mean it's tightly touching the nail? rot13(Be znlor V fubhyq or guvaxvat bhgfvqr gur obk, yvxr abg univat vg gbhpuvat gur anvyf va gur svefg cynpr be pbzvat hc jvgu fbzr jnl gb ynhapu vg) $\endgroup$ Commented Sep 25 at 6:18
  • $\begingroup$ @WeCanDoItGuys. 1) Well, that’s the question. 2) Yes, if it’s around only one nail, then it’s still stretched and tightly touching that nail (imagine the nails are very thick). It’s relaxed only when it’s not surrounding any nail. $\endgroup$ Commented Sep 25 at 6:23
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    $\begingroup$ Maybe a more rigorous way to say that the band relaxes is that it can be removed from the contraption without leaving the plane just above the board, say if a second board were placed atop the nails. $\endgroup$ Commented Sep 25 at 8:24
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    $\begingroup$ I can find two questions referencing this classic question, but neither of them actually ask it. The former does have an answer which answers this question. $\endgroup$ Commented Sep 25 at 8:31
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    $\begingroup$ There's also Connect three rings, though perhaps the equivalence isn't obvious to those not versed in topology. Likewise, A special kind of knot. $\endgroup$ Commented Sep 25 at 8:33

2 Answers 2

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Start by

applying the two-nail example to three nails. Removing the top nail works, but the two bottom nails are still connected.

simple loop around 3 nails on a board

To fix this,

both times we would pass under the two bottom nails, we should instead copy the two-nail pattern of crossing between them, doubling back under both, and crossing between them again.

animation of rubber band wrapped around 3 nails on a board

Final result:

complex loop around 3 nails on a board

To check that this works,

starting with the two-nail example, number the nails 1 to 2 from left to right. Starting from the top and traveling clockwise, record the (counter)clockwise orientation we travel around each nail: 1↻ 2↻ 1↺ 2↺. Removing the first nail results in 2↻ 2↺, which annihilates to nothing. Similarly, removing the second nail results in 1↻ 1↺, which also annihilates.

For our three-nail solution, number the nails 1 to 3 from the top clockwise and travel clockwise along the rubber band starting from the top, as in the animation: 1↻ 2↻ 3↺ 2↺ 3↻ 1↺ 3↺ 2↻ 3↻ 2↺. Whenever we consecutively wrap around the same nail in opposite directions, we can eliminate both.

  1. Removing 1: 2↻ 3↺ 2↺ 3↻ 3↺ 2↻ 3↻ 2↺ → 2↻ 3↺ 2↺ 2↻ 3↻ 2↺ → 2↻ 3↺ 3↻ 2↺ → 2↻ 2↺ → 0.
  2. Removing 2: 1↻ 3↺ 3↻ 1↺ 3↺ 3↻ → 1↻ 1↺ → 0.
  3. Removing 3: 1↻ 2↻ 2↺ 1↺ 2↻ 2↺ → 1↻ 1↺ → 0.

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  • $\begingroup$ Correct, this is exactly what I had in mind! That’s a cool animation btw. $\endgroup$ Commented Sep 25 at 13:14
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I know the solution has been posted already, but can someone explain to my dumb brain why the following isn't a valid solution?

PWOING!

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    $\begingroup$ Shouldn’t this be in a comment under the OP? And I explicitly said in the question that a relaxed state is when the band doesn’t surround any nail. In your final panel, it surrounds two nails. $\endgroup$ Commented Sep 25 at 14:00
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    $\begingroup$ Can't do images in comments, and this is essentially an attempt at an answer. Either way, that makes sense now, ty! $\endgroup$ Commented Sep 25 at 14:06
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    $\begingroup$ The puzzle gives bespoke definitions of the key terms: "The band is relaxed when it does not surround any nail, and it is stretched otherwise." It has nothing to do with tension on the band. By those definitions, it is clear that the band is "stretched" in all the images in this answer. $\endgroup$ Commented Sep 25 at 14:32
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    $\begingroup$ Here's a hack to add images in comments. Paste the image in an answer, copy the link created at the bottom, discard the answer, and paste the link in the comment. See this comment for instance. Incidentally, they ask the same question you did. $\endgroup$ Commented Sep 25 at 15:10
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    $\begingroup$ In other words, consider the rubber band of infinite elasticity - it will always shrink tighter around whatever's inside. It can't "relax" around even a single nail. $\endgroup$ Commented Sep 25 at 16:02

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