# Half filled water bottle

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?

• 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
Commented Aug 27, 2019 at 1:46
• No, but you can tell if it's half-empty. Commented Aug 28, 2019 at 1:36

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

• 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. Commented Aug 26, 2019 at 9:24
• The question doesn’t mention that you are given gravity. Commented Aug 26, 2019 at 20:01
• @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. Commented Aug 26, 2019 at 20:19
• @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-
Commented Aug 27, 2019 at 0:25
• 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. Commented Aug 27, 2019 at 9:03

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.

– Rubio
Commented Aug 27, 2019 at 1:39
• @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). Commented Aug 27, 2019 at 2:31
• I prefer this answer as it takes into account for symmetrically weird shaped bottles Commented Aug 27, 2019 at 2:31
• 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... Commented Aug 27, 2019 at 12:09
• 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. Commented Aug 27, 2019 at 15:33

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 $$(Y-Z) = 2 (X-Y)$$.

• Doesn't the question presume you're not going to remove the water in it? Commented Aug 27, 2019 at 12:27
• If you just emptied the bottle I can tell you without any math if the bottle is half filled. Commented Aug 28, 2019 at 15:45

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?)

• 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. Commented Aug 27, 2019 at 8:12
• 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. Commented Aug 27, 2019 at 14:47
• @Sanchises ok...yes ... I think you're presuming it's half full? ... the question presumes it's not. Commented Aug 27, 2019 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. Commented Aug 28, 2019 at 5:48
• @Sanchises Yep, you are indeed right. Commented Aug 28, 2019 at 14:48