Saturday, February 16, 2013

A Pair of Jugs

Time and Space
Part One
I'm sorry, but you knew there was going to be smut when you entered into this little student-teacher arrangement and this set of puzzles is too good an opportunity to miss.

Of course I'm speaking not about any part of the female anatomy, but one of the oldest and most varied puzzle sets there is - measuring puzzles. 

Usually packaged as water measuring puzzles these can also feature candle clocks, hour glasses and even dynamite fuses. The aim however is always the same, using two or more unmarked items of known quantity, work out how to make a further quantity. Let's break out those jugs for an example.

Two Jugs Puzzle - Fives and Threes

You are presented with two jugs, one of which has a volume of half a litre and another with a volume of 300 millilitres.

How can you use these two jugs to measure out a volume of 400 millilitres?

First things first, we want to convert the volumes of the jugs into nice easy to work with numbers. Here the decilitre - one tenth of a litre is the perfect unit of measurement to work in as it gives volumes of 5dl and 3dl for the jugs, and 4dl for the target.

There's nothing to stop us working in a completely made up unit either, if it makes the numbers easier. The puzzle is all about ratios rather than the numbers themselves and it's the processes we go through that matter, not the numbers.

So we've simplified our problem down to '5 and 3, get  4' which is much easier to work with, next we need a method of solution.

Now, the thing is, there is actually a general method for solving these kinds of problems and indeed for deciding whether any combination is possible to be solved. Unfortunately if you learned the method, you'd take most of the fun out of solving the puzzles - and there are many, many jugs puzzles. So in this case, I'd avoid learning it.

On the other hand if you are of a mathematical persuasion, it is a reasonably interesting exercise to work out the rule that governs which of these kinds of puzzle are solvable, so you might want to skip the rest of this post and have a play around. I suggest finding out what happens when the two jugs are the same, or multiples of each other and working from there. Either way, we'll be looking at several variation over the next couple of posts.

If, on the other hand, you just want to see the general method, or check your own, you can find it here.

So back to our puzzle, we need to manipulate the liquid in our vessels to give volumes other than 5 units and 3 units so clearly we have to pass liquid from one jug to the other leaving us with two quantities of liquid that are both known.

In this case we can fill 5, transfer into 3 and that leaves us with a 5 unit jug with 2 units inside and a 3 unit jug with 3 units inside. This is much easier to see as a diagram.

Unfortunately this doesn't actually get us anywhere towards getting a volume of four. These are, however, the first two steps towards getting a volume of two, the next step should be obvious. They're also the first two steps on the way to getting a volume of seven, There are three more steps, can yu see them?

Our other option therefore is to fill in the opposite direction start with the three and pour it into the five. It might not look like it does anything useful, but what's important is not what what we have filled, but what we have empty.

You should be able to see that in step four, we used the empty space in the five unit jug as two-unit measure, allowing us to split a three into a two and a one. Now we're just three steps away from the solution. These four steps are the answer to getting six units, and as you can one unit is also only a fifth step away. As it happens, four units takes the most steps of any number when using three and five unit jugs.

There you have it, the last key move is the transferal of one unit from the three unit just to the five unit jug. This, while not changing the volume we had, did change the empty spaces we had available to use, from a five and a two unit, to a four and a three unit. The final step was using the three unit space to reach our target.

Take some time to play around with this puzzle and make sure you understand the methods. You should be able to make up any number from one to eight units from these jugs.

Two Jugs Problem - Fives and Sevens

This time your jugs are five units and seven units, I wont bother to hide behind complicated volume measurements as this is a tough one.

Your task is to find the thirteen step process which gives one unit and the seven step process which gives eight units.

The two processes are examples of the two basic algorithms that power the solutions of this puzzle. From those processes you should be able to solve for all target volumes from 1 to 11, to solve 12 you'll need a third, trivial solution.

Have some fun with this as there'll be more jug puzzles in the coming weeks, and here's something to chill you to the bone - I have started studying cryptic crosswords ready for the spring!

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