In this festive series I aim to introduce you to Logic and Lateral Thinking puzzles by getting you to accept the three certainties of puzzling. I have written a little story about each of these truths and I call them the Puzzling Parables.
Welcome back, I hope you’re all feeling suitably refreshed from your Christmas break. We’ve still got a few puzzles to solve before twenty-twelve is out; not to mention the whole arena of mathematics to explore next year.
Over the New Year I have three puzzling parables to share with you, three little morals that will help you when you come to tackle all kinds of puzzles. Today we tackle what may be the most difficult task for any solver of puzzles, deciding what's important.
When we looked at riddles we learned how to divide them into questions, conditions, flavour and misdirection. These things aren’t unique to riddles of course, here has been plenty of misdirection in our exploration of word puzzles, and I try to put many of my puzzles into a context, giving them flavour.
Breaking down puzzles is an important skill; it is only by analysing what is really being asked in a puzzle that we can hope to discover a method and find a solution. There is another aspect to breaking down a puzzle however, and that is working out what is important.
Puzzle writers often have a tough time when polishing a puzzle ready for publishing. We might be devious and evil little souls, but there are most definitely rules of engagement, and we have to stick to them.
You can use this to your advantage, find the parts of a puzzle that are written in an awkward turn of phrase or seem out of place. Analyse each bit of information you have been given, and try to work out why that particular fact or condition has been set.
Here’s a nice festive example, where breaking down the puzzle leads to an answer.
Mrs Claus has cooked twenty-seven lovely Christmas puddings for Santa and the elves to enjoy as part of their job-well-done Christmas dinner back at the North Pole. A stanch traditionalist, she has hidden a single silver six-pence in in one of the puddings and mixed them up well.
Unfortunately she’s just had a message from elf and safety, to tell her that had she done the proper risk assessment she would know that elves are allergic to silver.
Rather than throw the puddings out Mrs Claus wants to work out which pudding contains the silver, and give that one to Santa. She knows that the pudding with the coin will be a little heavier than the others, but not heavy enough to feel by touch – so she’ll have to use scales.
She has a rickety of set of balance scales, the kind with a pan on each end of a bar, but she doesn’t have any weights for it. So all she can do is compare the weights of the puddings with each other.
To top it all off, the scales are only going to stand up to being used three more times before they collapse completely.
Is there are way she can identify the silver laden pudding in just three weighings?
Short answer: Yes
In fact the answer to this kind of puzzle is very, very rarely ‘no’ and where ‘no’ is the answer, there should be a clear way of proving it cannot be done. Of course you can't just say 'yes' and get away with it, you have to be able to prove it can be done.
There is a lot of favour in this puzzle, giving it its festive flavour – if you’ll excuse the repetition. If we strip away the flavour to simplify this puzzle, we’ll soon find that it’s one of a pretty famous collection of weighing puzzles that are often repackaged in new puzzle books.
26 Identical Items and One Heavier Item
One Heavier Item
There, that’s all a bit easier to follow now. Some things should be standing out like a sore thumb to your now seasoned puzzling brain and it’s these key facts that are going to help us formulate a solution.
Funny number 27, it’s certainly not an arbitrary choice like 10 or 100; indeed I’d go so far as saying that the fact that is such an odd number suggests that it’s the largest possible number of items that can always be sorted through with these conditions. Not to mention it can be divided nicely into 3, another number in the puzzle.
Not such an odd number here, but the 3 does tell us something useful. The most efficient way of sorting through those 27 items in stages it to eliminate the same proportion of the items each time. We need to end up with one item at the end, and this requires us to get rid of two thirds of the items at each stage.
27 > 9 > 3 > 1
This is the most important fact here, as it makes the puzzle solvable. The key thing to note is that ‘heavier’ is not the same as ‘different weight’, especially not with an old fashioned balance scale. ‘Heavier’ gives us a direction in which we expect the pans of the scale to move, which means if we weigh the same number of items on each pan, we can always discard the pan that raises up, as the pan that falls down has the heavier item.
Putting it All Together
So, if we’re working in thirds it stands to reason that we should start by dividing our 27 puddings into 3 nines and find a way to discard 18 of them.
Put nine on each side of the balance and observe what happens. If the left hand side goes down we know that the silver pudding is contained within the left had group, and we can discard the right hand group, and the group we didn’t weigh.
The reverse of course is also true, if the right hand group is heavier we discard both the left hand group, and the unweighed puddings.
The other possible situation is that the scales remain level. This of course tells us that the weightier pudding is in the unweighed group, and we should work with those next, discarding everything on the scales as safe for elves.
Now we have nine pudding we repeat the process in groups of three, finally in groups of one making a total of three weighings and leaving us with the single heavier pudding.
Here's a diagram showing how this works, no matter where the pudding sits, we can always identify it. The red arrow shows the group containing the weightier pudding at each stage.
To make sure you're got the grasp of this, try and work out what would happen if you had 28 puddings. Is there a number of puddings where it becomes impossible to identify the correct pudding at least some of time?
Merry Christmas Puzzlers, more fun and games in the New Year!