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You, a right cylinder glass of water, 3 buttons, instructions, and a large flat land that seems to go on forever, are the only things that exist in a universe. In this weird universe, the exact same physics apply. The 3 buttons are labeled "Less", "Half", and "More". The instructions are as follows:

Hello fellow human, I have abducted you from your puny planet. I am done testing you, and have wiped your memory. You have just 1 chance of escape. You must tell if the glass is half full, less than half, or more than half with water. If you are wrong I will remove the ground and let you fall into an abyss forever. I labeled 3 buttons accordingly. I have also given you perfect sight and perfect balance/handling. Please note perfect sight only helps you with levelling; it does not help in originally finding half. Also you do not have any measuring instrument and you cannot spill out or add anything. Good luck... Not.

The glass appears to be half full, but how can you tell for sure?

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    $\begingroup$ How about emptying the glass and pressing "less"? $\endgroup$
    – twisteroid ambassador
    Commented Oct 26, 2014 at 22:22
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    $\begingroup$ Half full of what, water, or matter? It is presumably the rest of the way full with air, so are we assured it isn't a trick question? Are we allowed to manipulate the glass? Are we allowed to remove water from the glass? $\endgroup$
    – doppelgreener
    Commented Oct 27, 2014 at 5:29
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    $\begingroup$ Refuse to participate, instead making a short but impassioned speech about experimental ethics, a la Star Trek. $\endgroup$
    – A E
    Commented Oct 27, 2014 at 14:20
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    $\begingroup$ Why do the instructions refer to you as a fellow human if he abducted you from your puny planet $\endgroup$
    – ediblecode
    Commented Oct 27, 2014 at 14:42
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    $\begingroup$ What is a right cylinder? Can such a cylinder be so heavy that you cannot lift it? $\endgroup$
    – Ole Tange
    Commented Oct 28, 2014 at 20:51

7 Answers 7

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Tilt the glass until the edge of the water is just touching the rim. Then see whether the highest point of the bottom of the glass is under, over or on the same level as the water.

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    $\begingroup$ Surface tension also complicates water. Though you probably don't know what that is, given the memory wipe. $\endgroup$
    – Chris Martin
    Commented Oct 27, 2014 at 4:54
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    $\begingroup$ I read "your memory has been wiped" as referring to "your memory of the tests we have been doing on you", otherwise you're unlikely to remember enough to even read the instructions. $\endgroup$
    – IanF1
    Commented Oct 27, 2014 at 12:11
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    $\begingroup$ Surface tension is a good point, but an attempt could be made to take the miniscus into account by eye. $\endgroup$
    – IanF1
    Commented Oct 27, 2014 at 12:12
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    $\begingroup$ I took "exact same physics" to mean "... as you're used to on the surface of your puny planet". Otherwise the problem wouldn't be well defined. $\endgroup$
    – IanF1
    Commented Oct 27, 2014 at 12:14
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    $\begingroup$ Cuts or holes would stop the glass from being "a right cylinder". $\endgroup$
    – IanF1
    Commented Oct 27, 2014 at 12:14
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After checking that I can breathe in the experiment room, I can confidently assert that the glass is more than half-full, either with water or air.

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    $\begingroup$ This is the only answer for this kind of challenges. $\endgroup$
    – Ismael Miguel
    Commented Oct 26, 2014 at 23:56
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    $\begingroup$ what-if.xkcd.com/6 $\endgroup$
    – Justin
    Commented Oct 27, 2014 at 4:07
  • $\begingroup$ Alternately, completely full of water and air. $\endgroup$
    – mccainz
    Commented Oct 27, 2014 at 19:29
  • $\begingroup$ What if it is exactly half-full? $\endgroup$
    – apnorton
    Commented Oct 28, 2014 at 16:07
  • $\begingroup$ @anorton The water and air together fill up the glass more than half-full. $\endgroup$
    – blakeoft
    Commented Jan 23, 2015 at 20:02
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Since the same laws of physics apply, I would expect capillary action to produce a concave meniscus at the edge where the water surface touches the glass. If the glass is very nearly half-filled, the tilting method proposed by user2021 would not be able to account for this effect.

The only way to be sure is to WAIT, since there is no time limit, and let evaporation takes place till you are confident enough to answer LESS.

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    $\begingroup$ Good way of dealing with the couple-of-atoms-short situation. But you might get thirsty by all that waiting - hey, there is half a cylinder of water sitting right there... $\endgroup$
    – Ole Tange
    Commented Oct 27, 2014 at 8:36
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    $\begingroup$ Who says evaporation even happens in this locale? $\endgroup$
    – Carl Witthoft
    Commented Oct 27, 2014 at 12:06
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    $\begingroup$ @Carl "In this weird universe, the exact same physics apply." $\endgroup$
    – David Conrad
    Commented Oct 27, 2014 at 18:30
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    $\begingroup$ OTOH, I'm not sure what gravity looks like when you're on an infinite plane, presumably of finite thickness (or gravity would be infinite for sure!). So I don't know how to evaluate "same physics apply" $\endgroup$
    – Carl Witthoft
    Commented Oct 27, 2014 at 20:38
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    $\begingroup$ @CarlWitthoft on an infinite plane the gravity would not fall off with 1/r^2 and would be constant orthogonal to the plane. This also means that no spaceship can ever escape this plane and any object that exists in this universe will come crashing down really hard sooner or later $\endgroup$
    – DenDenDo
    Commented Oct 28, 2014 at 15:46
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We are dealing with a power that can abduct you from your planet, wipe your memory and create a highly specialized environment. These powers clearly have access to technology that is superior to the state of the art of our technology. That means that this power will most likely be much better at measuring than your puny eyes, especially if they are combined with your human hands. Even human technology is better than that.

I would find it likely that the power could make the cylinder half full except for a couple of atoms. In which case you might be tempted to press "half" when the correct answer is "less".

That makes user2021's answer not strictly correct (or the puzzle malformed).

Unless someone comes up with an answer that deals with the couple-of-atoms-short situation, then the correct answer is: No, you cannot tell for sure in all situations.

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  • $\begingroup$ I was trying to add a story to a known puzzle, I've edited the instructions to accommodate for this. $\endgroup$
    – warspyking
    Commented Oct 26, 2014 at 22:55
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Since you have the instructions, take the instructions and place them against the opening of the cup.

Then, tilt the cup until it is horizontal.

Look at the cup from the side so what you are looking at looks like a circle with the water level acting like a chord.

Having perfect eyesight, determine if the chord is a diameter, using your perfect balance, balance the water so that it is perfectly flat and check to see that the line drawn on the bottom of the cup is a diameter. If it is, it is half full. If it is below that diameter (the diameter will be parallel to the flat land given that physics calls for gravity), then the cup is less than half full. If it the water level line is drawn above the diameter then it is more than half full.

No adding or removing of the liquid.

The liberty I am taking is that the instructions are perfectly flat and does not absorb water. To specify it even more, the instructions act as a "second bottom", i.e. a top, for the right cylinder.

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It's a trick question: The glas is full (around half would be water, and the remainder would be air).

For the puzzle to be precise in the line of the other answers, it'd need to explain that the question is 'to what degree is the glass filled with water', and explain that the assumed air contains no water vapor, nor is the ionisation of water taken into account (granted, that is very minimal, but I'm assuming aliens to be evil, so better safe then sorry).

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Close over cylinder with a flat surface, using your perfect handling this should be possible with a hand, or arm perhaps (assuming this is possible if accepted answer is possible), and slowly turn it 180 degrees, clockwise.

Check, with perfect eyesight, that there is a dry spot on the right side of the cylinder. If there is, when you are done, then less than half of it is full. If not - half. If before you reach 180 degrees the water meets a wet spot, it is more than half full.

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