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If you are given a marker and a transparent water bottle partially filled with water, can you tell if the bottle is half filled or not?

Akanksha
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    There's been a lot of commentary on this question (particularly on its accepted answer) which, I think, goes strongly to showing how very short posts need to be very careful in how they are worded. The current version is much better than what we started with. I'd caution askers to take care with short posts to ensure you say what you mean, and no more and no less; and equally caution answerers that for short posts by new users, please have some patience with inexact wording. Helping a new user find the right way to phrase their puzzle is good. Saying "-1" because of errant wording may not be. – Rubio Aug 27 '19 at 01:46
  • No, but you can tell if it's half-empty. – Khuldraeseth na'Barya Aug 28 '19 at 01:36

4 Answers4

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Make a mark at the water level and then turn the bottle upside-down. The water level should be the same as the mark.

JMP
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    I don't think this will work if you have a bottle that is shaped like a $\Psi$ for example, because turning it upside down will give three different water levels. – Reinier Aug 26 '19 at 09:24
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    The question doesn’t mention that you are given gravity. – dessert Aug 26 '19 at 20:01
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    @dessert it doesn't specify it takes place in a universe with the electromagnetic force and Pauli exclusion principle either, or that you're not trying to perform your experiments while riding a roller coaster. I don't think anybody was actually confused about any of these things. – Sarah G Aug 26 '19 at 20:19
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    @Reinier, even a psi shaped bottle could work, as long as the bottle is symmetric in some way (even rotational: if you had a bottle like an S-tetris block, this would still work). Just angle it so the line of symmetry is perpendicular to gravity. – BM- Aug 27 '19 at 00:25
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    Many comments regarding an older version of this puzzle and its wording have been moved to chat. – Rubio Aug 27 '19 at 01:47
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    The shape of the bottle is irrelevant. Half full refers to the volume. If half of the space is filled, then the line will match the water level at either orientation. You don't need any symmetry in the bottle, just for the flip to be exactly 180 degrees. – WittierDinosaur Aug 27 '19 at 09:03
  • I downvoted this before realising that it is self-evidently true! Aplologies, and upvoted. – Strawberry Aug 27 '19 at 13:28
  • I needed @WittierDinosaur to show me why this is the correct answer! – Auspex Aug 27 '19 at 15:34
  • @Reinier - have searched around and not located a bottle with a strange shape that you show! But even that would work when placed on its side! – Tim Aug 28 '19 at 06:58
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    @WittierDinosaur The symmetry of the bottle would more relate to trying to get the volume markings to be equal. In the psi example, on one end, all three "branches" of the psi are communicating vessels, so water level would be the same in each branch. When flipped, water cannot flow between the branches, and one branch could have a different level than the others. In that case, the marker lines wouldn't match water levels between the two scenarios, even if it was still half volume in each orientation. – JMac Aug 28 '19 at 14:39
  • Of course this is true...when you originally have the bottle upright, the water takes up a certain amount of volume, and the rest of the bottle is filled with air. Let's call the volume the water takes up $v$, the total volume of the bottle $1$, and the volume taken up by the air $1-v$. What does it mean for the bottle to be "half-filled"? It means that the volume taken up by the water is equal to the volume taken up by the air. Let's make the mark and turn the bottle upside down. – joshuaronis Sep 03 '19 at 13:45
  • The key thing to realize is that if the bottle is half full, the volume that the water didn't take up (the volume filled with air) prior to being turned around is equal to the volume it did take up. Now, once the bottle is turned around, the water will try to fill everywhere where there wasn't water before. If the bottle is originally more than half-full, it'll fill it, and there will still be some water left in the original half where the water was. If the bottle is less than half full, then it won't be able to fill that space. But if the bottle was exactly half full to begin... – joshuaronis Sep 03 '19 at 13:57
  • ...with, then it'll fill up the half it didn't fill up before, where there was air before, and there will be air everywhere where there was water before. So obviously it'll fill up to the "halfway point" where we made the mark.
    (Sorry for the extended comment - I could've definitely shortened this to a single comment. I didn't realize it didn't let me edit after I had finished writing)     – joshuaronis Sep 03 '19 at 13:58
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Lay the bottle on its side, mark the water level on both sides, then roll it 180 degrees.. if the water meets the lines, it is half full, under the lines - less than half, higher than the lines - more than half.

JMP
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frank
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    Welcome to Puzzling! (Take the [Tour]!) As a general suggestion: if you're posting an answer that's largely the same as an existing one, acknowledge the prior answer and indicate how yours differs, improves upon, or adds relevant detail to the answer already provided. That way there's no confusion about whether or not you just didn't see the earlier answer. And, of course, if you can't really explain in that manner why your answer is not in essence a simple duplicate, that's probably a good sign that your answer isn't adding anything to what's already been said, and shouldn't be posted. – Rubio Aug 27 '19 at 01:39
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    @Rubio agreed that this answer needs more detail to explain why it's better than the accepted answer. For the record, I'd argue that this is different and better that the currently accepted one, since the bottle is more likely to have a horizontal axis of symmetry when laid on its side. It would be even better if it were explicit about the importance of symmetry (like BM's comment on the other answer). – Ergwun Aug 27 '19 at 02:31
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    I prefer this answer as it takes into account for symmetrically weird shaped bottles – RozzA Aug 27 '19 at 02:31
  • yes, this one is correct first rotate the bottle sideways, in the way the bottle opening is horizontal... then mark the water level and finally turn it 180 and match the water level... – Sayed Mohd Ali Aug 27 '19 at 12:09
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    Except that WittierDinosaur is absolutely correct—symmetry is not necessary. If the bottle is half full, then there is as much air as water. If you flip it 180 degrees, even if it's not symmetric, the air should exactly fill the space the water originally filled. I only realized this because of his comment. – Auspex Aug 27 '19 at 15:33
  • This one is better than the accepted answer because bottles (well, usually!) are axially symmetric, but not symmetric otherwise. So, this solution will work much more reliably. – Damon Aug 28 '19 at 08:18
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Inspired by Randy and his assumption of scales and more water (but not of any symmetry to the bottle), here's a solution that doesn't require a freezer or a bottle of negligible weight:

Weigh the bottle with the water in it to get the weight of bottle and water, $X$. Fill the bottle with additional water and weigh again, to get the weight of a full bottle (including water), $Y$. Empty the bottle entirely and weigh it again to get the weight of the bottle alone, $Z$. The bottle is half filled if $(Y-Z) = 2 (X-Y)$.

Ergwun
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I'm going to assume the bottle is any shape...that is, no particular shape that happens to have some symmetry.

Therefore we can't (easily) use 'observational geometry' to transform the bottle to indicate the same water level at different positions.

So I'm left with inferring the volume.

I freeze the bottle until the water is frozen (leaving the ability to pour in more water). I fill the bottle with cold water so the frozen water doesn't melt. I pour out the water, measure it's weight, and compare to the weight of the bottle.

(Assuming the transparent water bottle is of negligible weight. What's the solution if the bottle weighs alot?)

Randy Zeitman
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    Frozen water (ice) is less dense than cold water. You may have established that the same weights of ice and water occupied the bottle but that means that the ice, at the time, was occupying more than half of the volume. – Damien_The_Unbeliever Aug 27 '19 at 08:12
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    Therefore we can't (easily) use 'observational geometry' to transform the bottle to indicate the same water level at different positions. You could simply mark the water level around the perimeter of the bottle (even three dots will do). If you flip the bottle, you can find a new position that exactly matches the marked line iff the bottle is half full. – Sanchises Aug 27 '19 at 14:47
  • @Sanchises ok...yes ... I think you're presuming it's half full? ... the question presumes it's not. – Randy Zeitman Aug 27 '19 at 19:15
  • @Randy I make no presumption either way, I only say that this procedure (three dots and a flip) is only successful if the bottle is half full, and if you don't manage, it's not half full. – Sanchises Aug 28 '19 at 05:48
  • @Sanchises Yep, you are indeed right. – Randy Zeitman Aug 28 '19 at 14:48