Of power-cuts and coffteas!
1) We start with a simple one - You have 10 pieces( not pairs) each of black and white coloured socks in a drawer. Late one night, during a power-cut (Assume you are in India:-) , you reach into the drawer to fetch yourself a pair ( both pieces being of the same colour of course). How many pieces of sock do to you need to pull out before you can be certain you have a pair ? Essentially you have to pull out a certain number of socks without being able to note the colour. You then turn on a torch, and pick out a pair. What is the minimum number you need to pull out to ensure you have a pair? Assume a sock of the same shape can be used for both the right and the left foot. How about if we had 50 pieces of sock, with 10 each in 5 different colours ?
2) You have two glasses, one containing tea and another, coffee. Both glasses are filled to the same level, but not to the brim. You take one spoon of tea and pour it into coffee. Having mixed the solution well, you transfer one spoon of that back into the glass containing tea. Now, is there more tea in coffee, or more coffee in tea ?
Do post the solutions in the comments section if you've solved them( even if you're not sure you're right ;-)
PS - One of my friends told me I ought to add more pictures in my blog. So having taken his advice, here is a hot steaming cup of coffee for you ! :-) I'll try do a better job of choosing good pics next time !
Answer to question 1 (3 and 6). I am working on Q2 though I have two easy answers. (a)Taste the drinks and let your taste buds give you an answer. (b) It is an ill posed question. ;)
ReplyDeleteRafiki - Thanks for posting the solution, and for being the first one to comment on this blog !:-) Yours answers are right ! Ha ha , those of us with highly sensitive tastebuds ( I don't qualify:-) could probably taste and tell :-)
ReplyDeleteA quick explanation - On a lucky day, you would need to pick only two socks. Both would be black, or both white ! On an unlucky day ( read typical day:-), the first two you pick would be of different colours, so you pick a third one to arrive at a pair. The pair would then be of the same colour as that of the third sock you picked. Similarly, you pick 6 socks in the second case to ensure atleast two are of the same colour !
A1) To generalize, (if you are the perennial victim of Murphy's Law) you need to pick one sock more than the number of colors.
ReplyDeleteA2) There's more tea in coffee than there is coffee in tea. Because you put one full spoon of tea in the glass of coffee but you put back less than a spoonful of coffee in the glass of tea.
Hmm, should be simple common sense I guess but I did couple of probably unnecessary calculations..Please excuse the complexity. I look forward to your simple solution :)
ReplyDeleteLets start with 100ml tea and 100ml coffee
Lets assume first we take out say 1 tsp, say, thats 25ml...
100ml tea -25ml tea = 75mltea
100mlcoffee+25mltea=125ml tea+coffee mix
That means 25/125 = .2ml tea in every ml of what was originally coffee.
Now lets calculate what 25ml of this mix contains from the compositon of 1ml of that mixture: Every 25 ml has .2*25=5ml tea .. and then 25-5=20 ml coffee
Now, if we transfer back 1 tsp (25 ml) of that mix back to tea:
75ml tea+ 5ml tea+ 20ml coffee = 100ml mix
ie. 80ml tea and 20ml coffee
ie. 20 ml coffee in 100ml mix.
ie. 0.2ml coffee in every ml of what was originally tea.
The quantities are the same!
There is as much coffee in tea as there is tea in coffee....
Nevertheless, Karthik, I'd prefer it if the cup of hot coffee you offered us here in the pic is 100% coffee :)
I think I got A2 wrong. As anonymous has rightly pointed out, there is as much tea in coffee as there is coffee in tea.
ReplyDeleteThe flaw in my reasoning was that while we are adding less than a spoonful of coffee to the glass of tea, we are also taking away part of the tea we had mixed earlier in the coffee.
A formal proof would be as follows:
Let's say the total volume of liquid in each glass at the start is 'X'.
Let the volume of liquid that can be taken in a spoon be 'S'. Let S be some fraction 'a' of 'X'.
So, S=aX
At the start,
Glass-1 has X ml of tea and 0 ml coffee
Glass-2 has 0 ml of tea and X ml coffee
After transferring a spoonful from glass-1 to glass-2,
Glass-1 has X-aX ml of tea and 0 ml coffee
Glass-2 has aX ml of tea and X ml coffee
If we now take a spoonful from glass-2, it will have (a^2)*X/(1+a) ml of tea and ax/(1+a) ml of coffee.
So, after transferring a spoonful from glass-2 to glass-1,
Glass-1 has X-aX+(a^2)*X/(1+a) ml of tea and ax/(1+a) ml coffee
Glass-2 has aX-a^2)*X/(1+a) ml of tea and X-ax/(1+a) ml coffee
Finally, calculating the ratios we have:
Proportion of coffee in glass-1 = a/(1+a)
Proportion of tea in glass-2 = a/(1+a)
Conclusion: There is as much tea in coffee as there is coffee in tea.
NOTE: This holds only if we define 'more' in terms of volume. If we define 'more' in terms of mass, then we'll have to consider the densities of tea and coffee and hence the solution becomes a little more involved!
Lalit,
ReplyDeleteThank you for sharing your comments !:-)
A1) Thanks ! Yes it would generalize the way you said!
A2) Actually most of us initially think the way you described. We are transferring a 'concentrated' spoon of tea into the coffee, but transferring a 'diluted' spoon of coffee back into the tea. So we think there is more tea in coffee.
Actually this is not true. Infact , because we are transferring diluted coffee back into the tea, we are bringing back some of the concentrated tea that we originally transferred !
Anonymous,
ReplyDeleteThank you for the excellent answer !!! You win ten points:-)
Lalit,
ReplyDeleteI just saw your new comment ! Thanks for posting the generalized answer !:-) You are right; there is as much tea in coffe as there is coffee in tea! Yes of course, it was meant in terms of volume. Using mass would make it unnecessarily complicated, and I chose this question because what happens is counter-intuitive to what we think !
Alternative proof through logic instead of math-
ReplyDeleteThe volume of liquid in both glasses is same before and after the transfer, since a spoon of liquid is transferred both ways. So any gap that was created by transferring tea into coffee obviously has to be filled by coffee to make up for the loss of volume ! If you 'effectively' transferred 'x' amount of tea into coffee, to keep the level of liquids same as before, 'x' amount of coffee would have to come back ! So as much tea in coffee as there is coffee in tea :-)
Anonymous - Yes, you can assume that the picture you see is one of 100% pure coffee ! :-) Maybe it inspired you to find the solution:-)
ReplyDeleteOh cool! nice solution (of tea and coffee and to the problem).
ReplyDeleteThank you Karthik for the points as well as the coffee :) Besh besh romba nanna irukku :D
ReplyDeleteAwesome intuitive, powerful and simple solution from you..
And great fundas and enthu everyone:) Keep it going!! :)