## Can You Solve The Diluted Wine Puzzle?

Hey, this is Presh Talwalkar. A servant has a method to steal wine. He removes three cups from a barrel of wine and replaces it with 3 cups of water. The next day he wants more wine. So he does the same thing. He removes three cups from the same barrel that now contains diluted wine and replaces it with 3 cups of water. The following day he repeats this one more time. So he has drawn a total of three times from the same barrel and has poured back a total of nine cups of water. At this point the barrel is 50% wine and fifty percent water. The question is how many cups of wine were originally in the barrel? This problem is a classic mathematical puzzle from the 16th century. Can you figure it out? Give this problem a try and when you’re ready keep watching the video for the solution. We’ll get started with the barrel of wine. We’re going to keep track of the number of cups of wine in the barrel and the concentration of wine in the barrel. At the beginning we’ll denote the unknown amount of wine by the variable x cups. The barrel at first contains only wine, so it’s concentration will be one or it’s a hundred percent wine. The servant then removes three cups from the barrel and replaces it with 3 cups of water. The barrel will have a diluted amount of wine. We know that three cups of wine are removed so the total amount of wine remaining will be x minus 3. Since the servant has placed back in three cups of water the total amount of liquid in the barrel remains the same at x. So the concentration of wine will be the amount of wine x minus 3 divided by the total volume of liquid x. If we divide this through by x we have 1 minus 3 over x. It will be convenient to write it in this form. The next day the servant does the same thing so the wine and the barrel gets diluted even more. How much wine will be in the barrel? When we already have x minus 3 cups that are in there and then he removes 3 cups of the diluted wine. So the amount of wine remaining will be x minus 3 minus 3 times the diluted wine that he is removing. The concentration of the diluted wine allows us to figure out the exact amount of wine that he’s removing as opposed to water. So we can now simplify this fraction to figure out the amount of wine that’s remaining in the barrel. This will be x minus 6 plus 9 over x. Once again the total volume of liquid remains at x because the servant is replacing the wine that he removes with water. So the concentration of wine will be this amount of wine divided by x. We can then divide through and then we can actually see that we can factor this term. We end up with the concentration of 1 minus 3 over x, the quantity squared. So the following day the servant does this one more time, so the wine gets even more diluted? So from the amount of wine in the previous day, we subtract out three times the concentration of wine from the previous day. So we can simplify this fraction to figure out the amount of wine that’s remaining in the barrel. This will be 9 minus x plus 27 over x minus 27 over x squared. This amount divided by the total volume x is how much of the wine concentration is remaining in the barrel. I’ll go ahead and simplify this and it actually ends up being one minus 3 over x cubed. So that’s the final concentration of wine now. We’ve gone through all this algebra, and I just want to point out there was a shortcut we could have used if we had realized earlier. After we had gone through the first step of figuring out the concentration of wine, we could have noticed that the concentration going from one hundred percent to the first step is found by multiplying by 1 minus 3 divided by x. Now in order to get to the next concentration. We actually could have just multiplied by the same percentage to get to the next concentration. We could have done the same thing once more when we’re looking at the concentration and we’re iterating the same process the concentrations get multiplied by each other. In any case we figured out the concentration of wine at the very end. And we are given this is equal to fifty percent. So we now have an algebra problem to solve. We have the quantity 1 minus 3 over x cubed equal to one-half. We can take the cube root of each side, and then we can isolate and solve for the term x. We end up with x being equal to three times the cube root of 2 all divided by the cube root of 2 minus 1. And this gives us our answer that there were approximately 14.54 cups of wine in the barrel to start with. Did you figure this out? Thanks for watching this video! Please subscribe to my channel. I make videos on math and game theory. You can catch on my blog Mind Your Decisions that you can follow on Facebook, Google+, and Patreon. You can catch around social media @preshtalwalkar. And if you like this video, please check on my books their links in the video description

## Arul Senthu

Dec 12, 2017, 8:04 pmSimple solution.

Step 1: Setup a CCTV.

Step 2: Save the Footage.

Step 3: Handover it to police.

Apart from jokes it is really good explanation. Keep it up

## Hyperbolium

Dec 12, 2017, 3:42 amMy solution before watching the video: 3/(1-(1/2)^(1/3)) or 14.54 cups.

After watching the video: I have the correct answer, but it isn't simplified. Damn.

## Tech Master Pavit

Dec 12, 2017, 7:47 amToo easy……

Answer is14 cups app.

## hobo doc

Dec 12, 2017, 6:48 amThis is an exponential change problem. f(n) = abⁿ. In this situation, a = starting amount of wine (and the starting total of cups), f(n) = .5 * a, b = fraction at end of each ‘period’, and n = # of ‘periods’ which is 3. You can solve for b by setting up the equation—->. .5*a = a*b³ and b ≅ .7937; every ‘period’ there is a -20.63% change in the amount of wine from the previous ‘period’. After 1 ‘period’, a – 3 ≅ .7937 * a and a ≅ 14.54.

## Reaghan

Dec 12, 2017, 1:37 pmall i did is count how many things of water he puts back and i ended up with 15 and that's pretty close to 14.54!!!

## Therese Kesler

Jan 1, 2018, 9:39 pmthis is too mathy for me

## Rachel Landa

Jan 1, 2018, 4:55 amShouldn't the answer be 18?

## Google User

Jan 1, 2018, 2:44 amUsed recursive functions for this in c++, but the solution is easier on pencil and paper.

## Djms&capo DeMorais

Jan 1, 2018, 6:02 amYeah ,soon as the Punjabi pours the first glass of water inside , the second glass of wine he takes off is no longer wine…and so on.

## Mark Hurst

Jan 1, 2018, 7:40 pmif the 50%concentration contained 14.54 cups, then wouldn't you have started with 29.08 cups?

## 6string86

Jan 1, 2018, 2:41 pmI solved it but didn't have the confidence that this was the correct answer till I saw the solution 🙂

## MarbleSwan666

Jan 1, 2018, 3:35 amwait isnt it 18? pretend he took all 9 cups at once, and put 9 cups/water back. if its 50% wine now, its 18.

## dbltrplx

Jan 1, 2018, 8:22 pmI solved it by drinking all 3 cups of wine 🍷.

## Hugh MacDonald

Jan 1, 2018, 8:10 pmFor openers, the composition of alcohol, sugars, and water in wine varies, and wine is NOT homogenized, so this "solution" is imprecise. And, "…approximately 14.54 cups of wine…." is no real answer. It's an irrational number. And what's with the round bottomed barrels? Are you using an irrational cooper? Typical of the imprecise, pointless and loony crap on this site.

## Adrián Hernández

Feb 2, 2018, 2:39 amRed red wine, IT'S UP TO YOUUU!

## ZEDION

Feb 2, 2018, 8:12 am1: Cube root of 0.5 = 0.7937, this means after each day, the amount of wine left in the barrel is 0.7937 times what it was the day before (i.e. 0.7937 ^ 3 = 0.5), thus resulting in 50% remaining after 3 iterations.

2: After the first day, 0.7937 is left, which means 0.2063 was taken away (1 – 0.7937 = 0.2063), since this was 3 cups of wine, each cup's volume is therefore 0.2063 / 3 = 0.06877

3: Using this, we can figure out that the original (100%) contains 14.54 cups, because 1 / 0.06877 = 14.54

## Krishnaraj Thadesar

Feb 2, 2018, 9:46 pmThis is just not my piece of cake

## Feral Feline

Feb 2, 2018, 10:29 pmI decided since it is your video I'd just let you figure it out. No sense me doing your homework for you. I do recall doing this problem when I was in jr. high tho.

## R.

Feb 2, 2018, 5:15 amThe barrel is a paraboloid

## iamadooddood

Feb 2, 2018, 9:20 amI got 6 + 3∛4 + 3∛2 as the exact capacity.

## Frank Jiang

Feb 2, 2018, 3:28 pmi didnt learn algebra yet but i understnad it

## Mister Noodle

Feb 2, 2018, 10:08 amWhile this is one solution to the problem with a certain frame of mind, I do not believe it to be the only solution. There are a few variables that are being neglected, centered mainly around how the water dilutes the wine. Does the water completely evenly disperse in the wine considering the respective densities of water and wine, which are different? Additionally, if the servant has the method of stealing this wine, has this been done before to the wine barrel in question? The wine could already be diluted on the first day. This sounds closer to needing a differential equation than simple algebra, at least in my opinion.

## James Matthews

Feb 2, 2018, 12:24 amIf you've poured back 9 cups of water and the barrel is now half wine and half water, why isn't the answer 18 barrels?

## Music In A Can

Feb 2, 2018, 7:44 pmi actually thought it was 18, since he withdrawed 9 cups of wine, and now the barrel is 50% wine, so, that would mean that if the barrel were 100% wine there would be 18 cups inside.

## Koen Th

Feb 2, 2018, 10:15 amBefore watching: Suppose the barrel contains n glasses. The wine in it is diluted according to a geometric series, with a factor k=(n-3)/n each day. After 3 days half of the wine is left, so k^3=1/2, so 1/k=n/(n-3) is the cube root of 2 (which I call r here for notation ease). This solves as n=3r/(r-1).

## Retro Gaming – Clash Of Clans

Mar 3, 2018, 2:59 amI did the same thing basically but I kept track of – "The amount of wine removed" and "The concentration of water" for some reason lol

## larry deckert

Mar 3, 2018, 3:54 amneed a bigger barrel… thats good wine 🙂

## 式神• TrueAC •

Mar 3, 2018, 11:48 amJust buy your own wine…

## Dawid Zwiewka

Mar 3, 2018, 7:33 pmI calculated that it's ~14,541966305589217918556748739652 but really expected it will be some integer number instead.

## multiz0rak

Mar 3, 2018, 8:19 pmi just assumed x^3=0.5, x being the amount of dilution on every step. this gives x roughly equal to 0.8, ie 20% is 3 cups, hence the barrel is 15 cups.

## Fabian Haecker

Mar 3, 2018, 2:05 pmOr you just use differential equations

## Captain Levi

Mar 3, 2018, 3:53 pmDid you figure this out? of course i did!

## Daniel Goodrich

Mar 3, 2018, 10:24 pmStealing is bad M'kay

## StoryTimeUSA

Mar 3, 2018, 6:58 amevery time i think i figured it out, i didnt.

## Josef Ibrahim

Mar 3, 2018, 5:48 pmWooow I could do it! Methode:

x := amount of wine; w := wine; t := water, D := Day; after "//" is a comment

Idea: What happens everyday is that the old amount becomes [OldAmount * (1- (3/x)) + 3t]

So in D0 we have got x wine cups and then:

D1: xw * (1-(3/x)) + 3t

= (x-3)w + 3t //(x-3 cups of wine + 3 cups of water)

D2: [ ( (x-3)w + 3t * (1 – (3/x) ) ] + 3t //(= the old amount * (1-(3/x)) which means the old amout decreases 3 cups)

= xw – 6w + 6t + (9/x)w – (9/x)t

D3: [(xw – 6w + 6t + (9/x)w – (9/x)t) * (1 – (3/x) ) ] + 3t

= …. some math … = xw – 9w + 9t + (27/x)w – (27/x)t – (27/(x^2))w + (27/(x^2))t

So after the 3rd day the concentration percentage is 50% – 50% which means that the amount of w = amountof t:

x – 9 + 27/x – 27/x^2 = 9 – 27/x + 27/x^2

<=>

x^3 – 18x^2 + 54x – 54 = 0

And when solving this equation we get: x = 14.541966305 ~ 14.5 which means we had ~14.5 cups of pure wine Q.E.D

## Robin Katsu

Mar 3, 2018, 12:55 amhow to do without using algebra, i guessed it to be 15 he steals 3 and replaces it with 3 water so now there is 12 wine 3 water in the barrel thats 80% wine 20% water. day 2 he steals again 3 cups of 80% mix which is 2.4 cups of wine and 0.6 of water now there 15-3-2.4 = 9.6 cups in the barrel and that is 64% day 3 another theft of 3 cups mixed wine and water at 64% which is 1.92 wine and 1.08 water. now there 9.6-1.92=7.68 wine left and 7.32 water these 2 numbers should have been 7.5 if 15 was the right answer so then i tried 14 as answer which gave me less wine then water so it had to be between 14 and 15 i kept narrowing it down until i got the right answer

## IMZ3

Mar 3, 2018, 2:30 ami got 12 cups… pen and paper

## Fif0l

Mar 3, 2018, 11:49 amI don't know why he would stop at 3 days. If he kept doing this for 50 years he would produce a homeopathic cure for hangover.

## ronrooz

Mar 3, 2018, 1:24 pmI think the answer is wrong, as well as most comments. It is known that the barrel contains x glasses of wine to begin with and at then end it still contains the same amount. We know that we have added nine glasses of water. So if the amount is still x, nine glasses of wine (or mixed solution) must have been removed from x or the total amount would not be x anymore. So if nine glasses of water equal 50% of the solution, the total volume must be 18 glasses. This was the original amount.

This problem is often presented in an alternative way: you have one glass that is partly filled with water and one glass that has wine in it. You pour an unknown amount of water into the wine and then pour back the same amount of the mix into that glass that contains the water. Now you have two glasses that contain a mix. The question is: is there more water in the wine or is there more wine in the water?

## Jimmy Rosendahl

Mar 3, 2018, 8:45 pmX – 9 cups = 50%

X = 18 cups

## Friar Tuck

Mar 3, 2018, 6:23 amWhen the servant is sinning, no alcoholics are winning.

## daniel pardenilla

Mar 3, 2018, 12:10 pmi thought it was 18, since originally the barrel was 100% wine. But now it is now 50% wine, and 50% percent water. Therefore, he took 50% wine. He drank a total of 9 cups. Therefore, the 50% of the wine he tool was 9 cups. Since 50% is equal to 9 cups, the remaining 50% wine must also be 9 cups. When you add the wine he took which was 9 cups(50%), and the wine left which is also 9 cups(50%), the sum is 18 cups which is 100%, and 100% was the original amount of wine right?

## Lol Bro

Mar 3, 2018, 10:54 amIf he took 9 cups and the barrel is now 50/50 originally it had 18 cups of wine a harder question would be how much wine did the butler get

## Amy

Mar 3, 2018, 2:33 pmIt depends how big your cup is

## Wario64I

Mar 3, 2018, 5:40 pmI knew how to solve it I was just too lazy to actually count it

## futilitariano

Mar 3, 2018, 12:59 amAh shucks. I came up with the answer as a solution to the equation x^3 – 18x^2 + 54x – 54 = 0 and noted that the solution is not a rational number but found that the desired solution is close to 15.

## Erin S.

Mar 3, 2018, 4:59 amI get this route chosen to get the answer mathematically however, order of process should still apply. Diluted wine is still measured at one cup. The question wasn’t how much wine was left in the barrel as opposed to water it asked how many cups of wine were originally in the barrel. It’s a total mass equation a basic one at that not dilution percentage. The last sentence in first paragraph and all of the second paragraph made that clear. Which made all previous information unnecessary. I know your way was super smart and I agree with you mathematically however I believe this was a trick question for smartypants.

## Shaheer ziya

Mar 3, 2018, 4:16 pmQuit your wining about how y'all thought it was 18 cups, i too once thought that but it was naive of me.

## Fram Technical Eng

Mar 3, 2018, 7:55 pmStart

X cups of pure wine in the barrel

Assumption:

When barrel content is removed and replaced with water the new mix is homogenous in the entire volume of the barrel, i.e. no forming of wine and water layers.

Solving for the water content:

Day 1

Removed 3 cups pure wine, added 3 cups of water.

The whole volume of the barrel stays at X cups.

The total water content in the barel now is 3 cups.

Day 2

Removed 3 cups diluted wine, added 3 cups of water.

The whole volume of the barrel stays at X cups.

3/X parts of the total water content from day 1 was removed and 3 cups water were added.

The total water content in the barrel is:

(3 cups from day 1) – [3/X * (3 cups from day 1)] + (3 cups from day 2), or

3 cups – 9/X cups + 3 cups = (6-9/X) cups of water

The total water content in the barel now is (6-9/X) cups of water.

Day 3

Removed 3 cups more diluted wine, added 3 cups of water.

The whole volume of the barrel stays at X cups.

3/X parts of the total water content from day 2 was removed and 3 cups water were added.

The total water content in the barrel is:

(6-9/X) cups of water from day 2 – [ 3/X * (6-9/X) cups of water from day 2] + 3 cups, or

6 cups – (9/X)cups – (3/X * 6) cups + (3/X * 9/X) cups + 3 cups, or

9 cups – (9/X)cups – (18/X) cups + 27/X^2 cups, or

9 cups – (27/X)cups + 27/X^2 cups

However, we know, that on day 3 the total volume of the water is X/2 cups.

Then 9 cups – (27/X)cups + 27/X^2 cups = X/2 cups

9 – 27/X + 27/X^2 = X/2

18 – 54/X + 54/X^2 = X

18X^2 – 54X + 54 = X^3

X^3 – 18X^2 + 54X – 54 = 0

X = 14.541966

## Donal Trampf

Mar 3, 2018, 11:17 ami thought it out like this: The servant removes 9 cups of wine and replaces it with 3 cups of water three times. in the end, there was 50% water and 50% wine in the barrel. 3 cups 3 times is 9 cups in total, so he added 9 cups of water in total. if 9 cups is 50%, then there must be 9 cups of wine in the barrel too. 9+9 is 18 so i thought it was 18 cups of wine at the start. explain where i went wrong so i can understand XD

## Bing

Mar 3, 2018, 4:26 pm18 cups in total. It literally took 10 seconds :/

## Abdullah Wosaibi

Mar 3, 2018, 8:24 amI almost got there! Solving it, I assumed that he replaces every cup of wine he takes with water immediately (not take 3 wine

thenadd 3 water). Got ~13.49 cups in the end, ain't even mad## JakobHogn

Mar 3, 2018, 2:05 pmI used percentage and got between 14 and 15

## to whom it may concern

Mar 3, 2018, 3:00 pmi couldnt be arsed to work it out so did 3x3x3 x2 for 18 and guessed at 15 to allow for dilution. which is accurate enough for me since 14.54 rounded to the nearest whole number is 15

## Ⱥffābłe

Mar 3, 2018, 12:05 amI got this really quick, but because I'm doing a GCSE I'm conditioned to expect an integer so i spent 3-4 minutes trying to understand where i went wrong 😉

## dagarith

Mar 3, 2018, 4:16 amWouldnt this be 18. He takes out 9 cups of wine and puts 9 cups of water back in. The wine is 50% diluted or water. This implies there was double that in the barel which is 18. Im confused why we needed the algebra.

## Casper YC

Apr 4, 2018, 2:05 amA secondary (gcse) chemistry question in China.

## Aerobrake GD

Apr 4, 2018, 12:02 amby the way, x greater equal to zero

## Gary Back

Apr 4, 2018, 3:36 pmHalf life of wine = 3 days.

## Kyle Mart

Apr 4, 2018, 8:33 amOMG everyone. I spent over 2 hours trying to solve this, but I did it. I actually found the equation fairly early, but the tricky part was balancing out the equation which should have been simple, but it has been a long time since I've done anything like this and couldn't bring myself to give up. I feel so satisfied finally getting the answer.

((x – 3) ^ 3) / (x ^ 3) = (1 / 2)

(x – 3) ^ 3 = (1 / 2) * (x ^ 3)

x – 3 = (1 / 2) ^ (1 / 3) * x

-3 = ((1 / 2) ^ (1 / 3) – 1)x

x = -3 / ((1 / 2) ^ (1 / 3) – 1) <<—- YES!!! Finally!!!

Answer: 14.541966305589217

## variszarins

Apr 4, 2018, 10:30 pmis this algebra?

## CamXD man

May 5, 2018, 7:59 pm18 cups. it works try it.

## Michael L

May 5, 2018, 3:32 amHere is a brute force solution in python: https://repl.it/repls/SleepyGracefulQuadrant

## didiletyouknow

May 5, 2018, 12:05 amI solved it!! I got a cubic equation which brought me to the solution x=3(2+3√2+2^(2/3)) which is 14,54

## james parker

May 5, 2018, 12:13 amI calculated ahead and got 18. If you removed 3 cups 3 times that = 9. If that leaves the barrel to 50%wine then the other 50% will also be enough for another 9 cups altogether. 50 + 50 = 100% 9 + 9 = 18. So shouldn't there be 18 cups altogether of wine from the beginning?

## Ethan Haase

May 5, 2018, 2:37 amthis was way too overcomplicated

## N D

May 5, 2018, 6:06 pm51.2% is much better.

## Link

May 5, 2018, 6:13 am9 cups of water

50% water and 50% Wine in the end so now 9 cups of wine

9+9=18

## buddyfred96

May 5, 2018, 11:23 pmSome clues I used to just guess at the answer are (I didn't feel all mathematical geometrical cube root about it), 1) it had to be less than 18 cups of wine total in the barrel originally (2 x 9 = 18), 2) then he took out 3 cups of wine, then progressively less than 3 cups the two times after the first time so I guessed apx 2.5 cups out, then apx 1.8 cups (proportionally even less) and added the three amounts together equals 7.3 cups, so… 3) the total of the original amount of wine in the barrel could have been apx 14.6 cups, or close to it. So, he only got less than half a gallon of wine from a barrel with actually less than a gallon of wine to start with. But in the middle ages they used wine to help purify water anyway, so the diluted wine was probably ok.

## Faeem Shaikh

Jun 6, 2018, 12:15 amThe servant poured 9 cups of water so in the barrel there is 50% wine and 50% water so 9 cups of water is 50% of bell that means there is 9 cups of wine in the bell which is 50% so the hundred percent would be 18 cups of wine

## Faeem Shaikh

Jun 6, 2018, 12:18 amMy Bad I just woke up he did it 3 different time first time he got pure wine then The diluted stuff

## Billy C

Jun 6, 2018, 3:57 pmA:3

B:-4

A=B A>B A<B

## M R

Jun 6, 2018, 12:18 amquestion: why is the solution no as follows;

lets go with x = 14.54

Day 1 – 14.54 – 3 = 11.54 (wine content) : 11.54/14.54 = .7936 (wine concentration)

Day 2 – 3 glasses (of diluted mixture) * .7936 = 2.381 (portion of wine content removed ) : 11.54 – 2.381 = 9.158 (wine content) : 9.156/14.54 = .6299 (wine concentration)

Day 3 – 3 glasses (of diluted mixture) * .6299 = 1.4998 (portion of wine content removed) : 9.1589 – 1.4998 = 7.6591 (wine content) : 7.6591/14.54 = .5267 (wine concentration)

What am I getting wrong?

## pauldzim

Jun 6, 2018, 9:22 amThat wine barrel looks suspiciously like a wine glass without the stem

## jamolargo afk

Jun 6, 2018, 5:34 pmso logically 50% of water means the other 50% is wine meaning 100% of wine = 18 cups of wine, but because the drink was diluted twice there should be less than 9 cups of wine in those 9 cups he took, which means somewhere along the way the amount of wine was reduced , because we can all agree that before we start the equation that there should be 9 cups of wine in that barrel and 9 cups of water, but half of 14.54 = 7.27 thus. that guy was robbed!

## Sean Yang

Jul 7, 2018, 4:14 pmI loved that solution, this problem took me awhile and 3 tries total; fascinating!

## XadadaX

Jul 7, 2018, 12:45 amWhy I had to try to solve this by myself. Why. Just why.

## Peter Sutherland

Jul 7, 2018, 5:16 amProlly

## M P

Jul 7, 2018, 2:21 pmThis is not a puzzle, but a problem of elementary algebra. A puzzle always contain a "think out of the box" shortcut, which is not the case here. Disappointing.

## maximiliano fontaine

Jul 7, 2018, 12:34 ami got 14 by mental calculous, close enough

## Sher Brooke

Aug 8, 2018, 1:02 amThe answer SHOULD be "18 cups of wine", not "~14.54 cups of wine". Why? Because the servant removes THREE cups of wine and replaces it with THREE cups of water, the next day, does the SAME thing, and does the SAME after that. So the servent removed NINE cups of wine, and the diluted wine is FIFTY% wine and FIFTY% water, so the barrel started with EIGHTEEN cups of wine. Not ~14.54.

## Devashish Jain

Aug 8, 2018, 11:32 pmSolved it in excel

## ijabo

Aug 8, 2018, 9:56 amThe real answer is that this guy is not being paid well since he can't afford his own wine and that that the guy who's paying him deserves to get his wine stolen

## Aditya Vikram Singh

Aug 8, 2018, 12:02 pmExcellent approach

## Lelouch Vi Britannia

Sep 9, 2018, 6:50 amYeah but there is an easier way to do this…Since the amount of wine replaced by water is the same as the amount of wine remaining, we can conclude that the initial amount of wine was 9*2=18 cups

Now don't wine about this

## Eytan Suchard

Sep 9, 2018, 12:52 pm(1-3/A)^3 = ((A-3)/A)^3 = 1/2, (A-3)/A = (1/2)^1/3, 3/(1-(1/2)^1/3)=A, A~=14.541966,

The pedagogically easy way to write the process is: A -> 3+(3+(3+(1-3/A))(1-3/A))(1-3/A)

## NumberblocksFan 64

Sep 9, 2018, 5:22 pmI solved it in less than 1 second

## Kaleb Bruwer

Oct 10, 2018, 9:58 pmI got 9.8086, but the Newton-Rapson took like 15 steps to start converging on something. This is a really unsatisfying answer for the time I put into it…

Edit: Well, I screwed up. Also, why am I doing math at 12pm? That's the real question.

## f4dy

Oct 10, 2018, 3:51 pmFastest way to solve this:

Assume 1 cup instead of 3 cups. Multiply by 3 in the end.

Tc=Total Cups

First day: Added 1 water cup, removed no water cups

Second day: Added 1 water cup, removed 1/Tc water cups

Third day: Added 1 water cup, removed (2/Tc – 1/Tc^2) water cups

By the third day, the total amount of water added = Tc/2

3 – 1/Tc – 2/Tc + 1/Tc^2 = Tc/2

Solve for 3Tc, you'll get 14.54

## QuasiELVIS

Oct 10, 2018, 10:04 amA shortcut would have been to realise it's similar to the formulas for compound interest.

## Federico Balzi

Oct 10, 2018, 4:56 amYou can solve this by logic too,

If you don’t know any complex math like me:

I started with saying that there are 10 cups in the barrell, to get some data.

By taking 3 cups, you remain with 70% or 7 cups of wine inside the barrel, we can go on and get another 30% out of this 7 and remain with 4.9 cups, one more time and we remain with 3.43~

Now that we got this number we can assume that whichever number of cups is in the barrell, will be the 34.3%

Because we need to make this number up to 50% we can start testing different numbers going upwards from (10) that we tested, till we get to 14.54, which is 14.54 * 0.343 (or 34%) = 15.54

## Marshy

Oct 10, 2018, 3:25 amI thought that it was 18 cups..

Because first now, we have a barrel 50% of water, 50% wine which means 1 half is water and the other half is wine.

Since there were 9 cups of water in there which make 50%, to make another 50% its 18 cups….

## Sigmund

Nov 11, 2018, 4:56 amsolved

## phil durre

Nov 11, 2018, 9:38 pmtook me some time, but cool puzzle

## sdfklsdfls

Dec 12, 2018, 10:43 pmCan't you just say 18 cups to start with? Doesn't that work?

## Mohammad Azadi

Dec 12, 2018, 12:46 amThanks a lot,but If question was how many cups so we better answer to corect number,ex.14 caps because my scale is cup same as solve a puzzel in natural nu. number .

## Orion D. Hunter

Feb 2, 2019, 8:52 pm18 cups

## Rajesh Mittal

Mar 3, 2019, 1:50 pmYou should have equate it with 50

## Rajesh Mittal

Mar 3, 2019, 1:50 pmIsn't it

## thisismycoolnickname

May 5, 2019, 10:36 pmWow he's got a huge cup.