# Unexpected Delay while Mixing Shots

You are hosting a party, and everything is going perfectly well, until you ask if anyone wants a kuulalaakeri shot. (In case you are not familiar with this highly treacherous Finnish beverage, it has just a little more than half of Jaloviina, and exactly the same amount of Punssi. Also, it's not at all uncommon for things to stop going perfectly well after a couple of them.)

This time however, the trouble hits even before you get to consume the shots: You have

• one full 50cl bottle of Jaloviina
• one full 50cl bottle of Punssi, and
• 5 empty shot glasses, of 5cl each

and you realise this looks an awful lot like a puzzle waiting to be invented.

So you challenge yourself to solve a puzzle with the following restrictions:

1. At no point can any drink be in any container apart from those mentioned
2. Whatever you put into one container immediately gets perfectly mixed with whatever was already there
3. At the end all the drinks must be in the two bottles (Oh, the poor guests..)

You figure out that a nice puzzle would be "What is the maximum amount of Jaloviina you can get into the Punssi bottle" under these restrictions. Can you solve it, or are you stuck until your guests get too impatient and attack the way too expensive cognac you were saving for a special occasion?

• "a little more than half of Jaloviina, and exactly the same amount of Punssi". I love drinks that give more than 100% – Dr Xorile Feb 14 at 19:11
• @DrXorile I told you the drink is highly treacherous, didn't I? :-) – Bass Feb 14 at 21:29
• @Bass, sä pahaa. (Tuleen Helsingistä.) – humn Feb 16 at 1:23
• So, is there a better solution than the given ones? – A. P. Feb 23 at 22:09
• If the correct intended answer has been provided, don't forget to $\color{green}{\text{Accept}}\,\color{green}{\checkmark}$ it! :) – Mr Pie May 16 at 22:58

I can get ...

33.33cl

Attained thus:

- Pour 5cl of Jaloviina into each of the 5 shot glasses. Jaloviina bottle is now half empty.
- Pour the Punssi bottle into the Jaloviina bottle to fill it.
- Reverse this - pour the Jaloviina bottle into the Punssi bottle to fill it.
- Repeat the back and forth until everything is well mixed. (16 back-and-forths should do...)
- At this point you have a 2:1 mixture of Punssi and Jaloviina in both bottles.
- Fill the Jaloviina bottle, leaving the Punssi bottle half full of 2:1 mix
(that is, it contains 16.67cl Punssi and 8.33cl Jaloviina).
- Pour the 5 shot glasses of pure Jaloviina into the Punssi bottle.
- The Punssi bottle now contains 8.33+(5*5)=33.33cl of Jaloviina (and 16.67cl Punssi).

• I don't think anyone will be able to do better than this. – Brandon_J Feb 14 at 20:21
• @Brandon_J Technically, you could alter this fractionally by first cooling both bottles to about 4°C, so that the liquid is at its most dense (50cl at room temperature becomes about 49.9cl at 4°C), then performing the above steps, pouring about 2cl (by eye) of the Jaloviina bottle into a shot glass, letting everything return to room temperature, then topping both bottles up. But that's a lot of effort to get an extra 0.2cl of Jaloviina into the Punssi bottle! – Chronocidal Feb 15 at 11:15
• @Chronocidal Or you just freeze half of the Punssi inside the shot glasses, fill the shot glasses with the rest and stack the ice cubes on top. Then the Punssi bottle is completely empty and you can fill 50 cl of Jaloviina into it. – A. P. Feb 15 at 18:15
• Yeah freezing the drinks is a nice way to get around this. – Brandon_J Feb 15 at 23:05

Initial guess:

You can remove 25 cl of stuff by putting it in those glasses => the highest ratio you can do is 50 : (50-25) ratio = 2:1. This amounts to 33 cl.

Math to the rescue:

You have X cl of Jaloviina in glasses + (50-X) : (50-25+X) ratio of 50-X cl mixture in bottle. Then you have: X + (50-X)^2 / (50-25+X+50-X) = 2500 - 25X + X^2 / 75.

You maximize this function by putting X to 0 or 25 (as you have it limited to 0 at minimum and 25 at maximum - you can't pour negative amounts for obvious reasons, while number of glasses limits you to 25 cl at maximum). So, either pour Punssi out, or pour Jaloviina out, mix well and pour shots to the correct bottle.

Lateral thinking:

Simply swap bottle labels. No mixing needed and you have full 50 cl of stuff in the correctly labeled bottle.

• I like the lateral thinking. – Matthew Barber Feb 15 at 1:31

I'd think that the ideal solution involves:

Having at least 25cl of Punssi in mid-air and the remainder in the shot glasses, so you can pour all the Jaloviina into the empty Punssi bottle, and eventually recapture the Punssi into the Jaloviina bottle.

As to how you can do this:

Set up a revolving cascade with the shot glasses, pouring from ever increasing heights and whipping empty glasses from top to bottom until the Punssi bottle is empty. You might need a few spare hands for this, but it'd make for an action-packed party game.

Alternatively...

Send both bottles into space. You can then have the entire contents floating around as bubbles in the microgravity, while you perform the transfers at your leisure.