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Here are a couple of nice puzzles.
Problem 1
Five glasses are stacked to a height of 34 cm. Two glasses are stacked to a height of 19 cm. What is the height of a single glass?
Problem 2
This is adapted from Alex Bellos’s puzzle column on The Guardian.
A wine bottle contains 750 cubic centimers of liquid. It has a height of 27 cm and a base diameter of 8 cm. It is shaped like a cylinder except for a dimple punt at the bottom and its shoulder and neck area at the top. The bottle is made of a thin plastic so its thickness can be ignored. The bottle on its base is filled with liquid, and the liquid measures 14 cm from the base to the middle. The bottle is turned upside down, and the liquid measures 19 cm from the neck to the middle section. What is the volume of the liquid?
As usual, watch the video for a solution.
Water Glass And Wine Bottle Riddles
Or keep reading.
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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
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Answer To Water Glass And Wine Bottle Riddles
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
Both puzzles are about solving a system of equations, but the second one is considerably harder. However, the second one does admit a trick which makes its solution very easy.
Problem 1
Let x be the height of the glass and h be the incremental height of a stacked glass.
The two and five glasses cases lead to the following equations:
x + h = 19
19 + 3h = 34
The second equation can be solved for h:
19 + 3h = 34
3h = 15
h = 5
Substituting into the first equation gives:
x + h = 19
x + 5 = 19
x = 14
The height of a single glass is 14 cm.
Problem 2
The bottle deviates from a perfect cylinder in two places. Let u be the deficient volume from the upper shoulder and neck and let b be the deficient volume from the bottom dimple punt.
A cylinder with a diameter of 8 has a radius of 4. With a height of 27, the cylinder’s volume is then:
πr2h
= π(42)(27)
= 432π
The wine bottle’s volume is 750 cubic cm, but it is also equal to the cylinder’s volume minus the deficient volumes. This gives the equation:
432π – u – b = 750
Let V be the volume of liquid. When the bottle is right side up, the volume is that of a cylinder with a radius of 4 and height of 14 minus the volume b. And when the bottle is upside down, it is the same volume of liquid, calculated as a cylinder with a radius of 4 and height of 19 minus the volume u. We get the equations:
V = π(42)(14) – b
V = 224π – b
V = π(42)(19) – u
V = 304π – u
Adding these equations gives:
2V = 528π – u – b
From the first equation:
432π – u – b = 750
–u – b = 750 – 432π
Substituting into the 2 volume equation gives:
2V = 528π – u – b
2V = 528π + 750 – 432π
2V = 96π + 750
V = 48π + 375 ≈ 525.80
The liquid has a volume of 48π + 375 ≈ 525.80 cubic centimeters.
Incredible solution!
This answer is interesting since it is equal to half the bottle 750/2 = 375 plus an amount of liquid equal to 3 cm in the middle (a cylinder with a radius of 4 and a height of 3). How is this the case?
There’s a remarkable trick to solve the puzzle with even fewer calculations required. The trick is to drink some of the liquid! The exact amount is to drink 3 cm. Here’s what happens in that case:
Notice the left bottle has 16 cm of air from the top to the middle, and the right bottle has the same amount 16 cm of liquid from the neck to the middle. The amount of air and liquid is unchanged between the two cases, and the sum of air and liquid is the volume of the bottle. The amount of air in the left is exactly the amount of liquid in the right. Therefore, the bottle is precisely half-full with liquid (or air) after drinking 3 cm of liquid!
So the volume of liquid is half the bottle, which is 750/2 = 375. We need to add back the 3 cm of liquid we drank, so we add a cylinder of radius 4 and height of 3 to get:
375 + π(4)2(3)
= 375 + 48π
≈ 525.80
What an incredible solution!
References
Water glass riddle on social media
(If you know the original author please let me know)
https://twitter.com/Math_World_/status/1660167837942423560
https://blaiselockhart.medium.com/how-tall-is-the-glass-e266e7ead74d601254
Alex Bellos Guardian Whiskey Puzzle
https://www.theguardian.com/science/2017/jan/16/did-you-solve-it-the-whisky-puzzle-that-will-have-you-on-the-rocks
https://www.theguardian.com/science/2017/jan/16/can-you-solve-it-the-whisky-puzzle-that-will-have-you-on-the-rocks
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