Mixing Tea and Coffee

EluZions.com Puzzles Logic : Paradox • Books

Start with a half cup of tea and a half cup of coffee. Take one tablespoon of the tea and mix it in with the coffee. Take one tablespoon of this mixture and mix it back in with the tea. Which of the two cups contains more of its original contents?

I've had more email about this puzzle than any other - perhaps even more than about anything else at EncycloZine! I hope the following will convince everyone. The argument seems crystal clear to me, so unless you can show me it's wrong, I'm not accepting any more explanations for there being more tea/coffee than coffee/tea. Please post your comments/explanations etc to our forum.

Imagine that by some magic, the tea added to the coffee floated to the top of the cup, and the coffee added to the tea floated to the top. Now if there was more tea in the tea cup than coffee in the coffee cup (or vice versa) then the dividing lines between liquids would be at different heights, as shown in the picture. Note that the total volume in each cup is the same, since we transferred equal volumes. Now, imagine transferring back the floating liquids - we'd end up with unequal amounts of tea and coffee, which is not what we started with! The only way to end up with equal volumes of both, is if the 'foreign' liquid in each cup is exactly the same.

Try looking at it this way: suppose the volume of tea is x, and the volume of coffee is x. Now you make the two transfers, and let's call the net amount of tea moved to the coffee cup t, and the net amount of coffee moved to the tea cup, c. So the volume of liquid in the tea cup is now x-t+c, and the volume of liquid in the coffee cup is x-c+t. But since equal volumes were moved, the volume of liquids in both cups is the same, i.e. x-t+c=x-c+t. So -t+c=-c+t, and then 2t=2c, i.e. t=c. The net volume of each liquid moved is the same, so neither cup contains more of its original content than the other.

Eric Bentley offers a similar explanation:

Here is another way to explain the Coffee and tea problem: Take 200 pennys and divide them into piles of 100 heads and 100 tails. Put 10 heads into the 100 tails pile. Now take ANY ten pennys from the 110 combination pile and put them back into the heads pile. For the heads pile you now have 90 heads plus any more heads which you brought back from the combination pile, plus any tails from the combination pile for a total of 100 pennys. The second pile will have the exact opposite composition.

Another easy way to test this is with a deck of cards separated into two piles of black and red. Take any quantity of red and mix with black. Shuffle the combination pile as much as you want. Select the original quantity of cards from the pile and put them back into the red pile. Now count the cards and you will find the number of red cards in one pile matches the number of black in the other.

Ozzy (?) says:

Ok, I have read the problem, and I have also read the follow up, and I still say it's wrong. Here is why.

Like in the last example in the follow up, lets say the cups are 100 oz each and a teaspoon is 10 oz. When you take 10 oz of tea and mix it with the coffee, you have 100 oz coffee and 10 oz tea. Once evenly mixed, total contents are 110 oz...for every 10 oz of coffee there is 1 oz of tea, which makes your ratio 10/1. So if you add a 10 oz sample of this mixture back into the tea, you are adding approximately 9.011111111 coffee/.999999999 tea. These numbers go on infinitly...it will never add perfectly. Now add this back to the tea. In the tea cup, you will have 90.999999999 tea and 9.011111111 coffee, which means for every 9.09 oz tea you have .901 oz coffee, making your ratio 9.09 tea/.901 coffee, while your coffee cup is still 10/1. I know this looks really confusing, but if you add it all up, the tea cup ends up with more of it's original content.

From Ashutosh:

I think this reasoning is wrong . What the puzzle asks for is to determine the contents of Tea and Coffee in eack cup after one round of transfer(tea into coffee cup and coffee into tea cup).

Here is what i think.
lets say both cups contain x parts of Tea and x parts of coffee.Noe I take One spoonful of tea and put into coffee cup, as a result it forms a mixture of tea and coffee in the coffee cup....now I take a spoonful of the mixture and put it back into the tea cup.As a result the contents of the tea cup now contain both tea and coffee as well.

It means the coffee cup would contain a little more of its ORIGINAL content as the mixture which was removed from the coffee cup would certainly contain a little less pure coffee
(as it would also contain some tea).

in short
the tea cup losses x parts of pure tea
the coffee cup in turn losses x-y parts of coffee(as y would correspond to some part of tea added earlier)..

what do you think?
Ashutosh

Eric Tillotson says:

After reviewing your coffee and tea puzzle and reading your explination, I found it a bit confusing, so I tried it logic it out again myself and heres what i came up with, This might help others to see it easier:
1) Starting with say 20 teaspoons each we have 20c and 20t.
2) taking 1 teaspoon of tea and adding it to the coffee we have 20c+1t and 19t
3) now take 1 teaspoon of the mix, that comes to (20c+1t)/21 because that is 1 teaspoon out of the now 21 teapoons of fluid. adding it back to the remaning 19t we have
4) 20c+1t-((20c+1t)/21) and 19t+((20c+1t)/21) multiply both parts by 21 and you get
5) 420c+21t-20c-1t and 399t+20c+1t or 400c+20t and 400t+20c producing equal perpostions in both cups.

Actually, I found that much more confusing, but, different strokes for different folks, I suppose... :)

Here's a nice clarification from Chuck Patterson:

Say both cups contain 100 oz, one has 100 oz of coffee and one has 100 oz of tea. You take 10 oz of tea and put it in the coffee cup. The coffee cup now has 100 oz of coffee and 10 oz of tea. the coffee and tea are now mixed. Now you take 10 oz of the mixture (which would be 9 oz of coffee 1 oz of tea) and put it in the tea. The resulting two cups would be TEA CUP containing 91 oz of tea and 9 oz of coffee. COFFEE CUP containing 91 oz of coffee and 9 oz of tea.

Bret says:

The concept of mixing and remixing may sound rather complicated, but its really not so long as you keep it at an elementary level. I remember an old math professor of mine telling us over and over "Don't try to make the problem more complicated than it is." Nonetheless after reading the reasons people thought the mixing problem didn't work, I wondered if it for some odd reason matters whether or not real values or integers were used. Nope! When I tried to verify the other posted explanations this is what I found:

1. After reading through Ozzy's explanation a couple times I was dumbfounded. I finally had to break out a calculator to follow what was going on, and low and behold, the numbers don't add up. I find that 10 times 10/11th's is 9.090909... not 9.0111111... Using 9.090909...oz coffee and 0.9090...oz tea on the remix, the numbers work out, and indeed the final mixtures are equal and opposite.

2. For Ashutosh's explanation, keep in mind when we are returning a mixture to the tea cup, it is at a lower level. Thus concentrations of the mixing element have more value, and once again, the problem balances out.

I hope this helps weed out some confusion. Is it disturbing that I love this stuff?

Happy Puzzling - Bret

Mikael says:

this is an easy explanation for the tea and cofee problem that so many people are finding hard to understand:

you start with the same amount of tea and cofee in each cup, no matter how much or how many times you mix them, as lobg as you end up with the same volume in each cup, you will always end up with a percentage of tea in 1 cup equal to the percentage of cofee in the other, lets say you have 85% tea and 15% cofee in you original tea cup, that means, the other 15% tea and 85% cofee HAS TO BE in the other cup.

you don't need a bunch of variables and formulas for this one, just some common sense!

Jason Bagge observes:

I'm new to this website, and think its great. This particular puzzle has been bothering me for some time though. Although I agree with your answer, there seems to be left a little problem of the formalue for this.

Below is my workings to provide a relation ship between liquid volume of each cup, and spoon. Note the different formulae at the end. This is what has made the puzzle so interesting: (Note worked example in brackets)

Let X = total volume of liquid in each cup , (assume 3)
C = coffee content
T = tea content
S = Spoon Volume, (assume 2)

Step 1. Transfer tablespoon of tea and drop into the coffee cup

TX -TS = CX+TS (volume of tea cup = X-S =1; volume of coffee cup =X+S =5)

Step 2. Mix the coffee and tea in the coffee cup.

Step 3. If we now take a table spoon of the mixture, the content of the spoon is

((CX+TS)/(X+S))S : ( (6/5)C +(4/5)T ) =2)

Step 4: We transfer the spoonful of mixture to the tea cup. Therefore:

TX-TS +((CX+TS)/(X+S))S= CX+TS - ((CX+TS)/(X+S))S

Step 5: If we now take the content of Coffee in the tea cup. (Take C's from LHS of formulae)

XS/(X+S), content of coffee = 3 x 2 /(3+2) = (1 1/5) out of 3

and the content of the Tea in the coffee cup. (Take T's from the RHS of the formulae)

S-S^2/(X+S) = 2-4/(3+2) = 2- 4/5 = 1 1/5 out of 3.

Hence they are the same even though the formulae is different for each side.

From gigliepuff:

Hi... In an attempt to prove the equality of the tea/coffee incorrect, i managed to prove it correct instead! And i think i have found the easiest way to explain. Since people find it hard to understand infinite numbers, why not work with numbers that are divisible?

So here it goes...

Start with 3 tsp of Tea and 3 tsp of Coffee. Move 1 tsp of tea into the coffee. There is now 3 tsp of coffee and 1 tsp of tea in the coffee cup. ie. 75% coffee and 25% tea in the mixture. Take out 1 tsp of the mixture and you are left with a 3 tsp mixture containing 75% coffee(0.75 x 3 tsp = 2.25 tsp) and 25% tea (0.25 x 3 tsp = 0.75 tsp)

When you add the mixture to the remaining 2 tsp of tea, you are adding 0.75 tsp of coffee and 0.25 tsp of tea.

So you have 0.75 tsp of coffee from the mixture and... 2.25 tsp of tea (2 tsp + 0.25 tsp).

And Voila!

From moriquendi:

hi, I enjoyed your problems and agree with your answer. All of the answers you have given from other people are very confusing. Your answer also has a flaw you start off with basing it on magic! THINK IN SIMPLER TERMS! Here's my solution:

1 cup tea and 1 cup coffee (dosen't matter how much) you remove 1 TBSP of tea and add it to the coffee. Since the properties of liquids is that they mix indefinatly, we can assume that they mix completely. So therefore, the 1 TBSP you remove from this mixture contains equal parts of tea and coffee so therefore you are adding back 1/2 TBSP of tea and adding 1/2 TBSP of coffee to the tea. that means only 1/2 TBSP has been removed from each.

I have several more like that, maybe I'll add them here. But, again, please don't email me - Please post your comments/explanations etc to our forum.