So, feels like the time has come to reveal my answer to the puzzle I posted a few months back.
But that would be much too simple and not wordy enough for me, so here’s a long and rambling explanation of how this puzzle works, or at least is meant to work…because the solution is really very simple indeed, it’s all about the journey on this one.
On the face of it, this is a riddle, plain and simple – a problem that needs to be tackled with some mixture of logic, intuition and insight.
Here’s the curious thing, though – I, erm, don’t love riddles. I respect them, I enjoy the cleverness of some of them, and I have no ‘problem’ with them. Well, no problem, that is, beyond the fact that my mind really struggles to solve them. For somebody who loves writing and words, I’m not terribly quick with clever wordplay, or methodical investigation.
My love and talents lie not with riddles, but with their distant, estranged and unpopular cousins – the trick question.
In preparing this post, I did a solid five minutes of research on what a ‘trick question’ actually is before giving up. This was a little more research than I needed to do, and also far less than I should to make this post rigorously sound. Anyway, the point is that it turns out that there is very little interest in what a trick question actually is…so, in absence of the wisdom of Wikipedia, allow me to present my own definition and examples.
A trick question, to my mind, is a question that relies upon misdirection and/or incorrect suppositions in order to lead the reader to a false conclusion. Often the answer is quite simple, but the question is phrased in such a way as to either produce (paradoxically) confusion or false clarity.
One of my favourite trick questions, for how simple it is, runs:
How many animals of each type is Moses said to have taken onto the ark?
Answer – none. Noah, on the other hand….
This trick question relies upon a few things to achieve its trick. Firstly, the simplicity of the question (making it easy to think of an answer immediately), coupled with the ‘generally known’ quality of the story of the ark and the flood. Also useful is the fact that both Moses and Noah are superficially similar Biblical figures, robed, bearded and popular, despite the fact that their stories are very different. In my own limited experience, it is a denominationally-agnostic question, equally likely to trick Christians and atheists alike.
That being said, it is also a very simple question, and there is a fair chance you got the right answer. So, let’s change gears with another trick question example.
Recently, I went to visit my Aunt Sally’s house. Aunt Sally lives exactly 4km away from me, and, as it was a pleasant day, I decided to walk – as long as I maintain a pace of 6km/hour, I would get there right on time, which is important – Aunt Sally is frighteningly fanatical when it comes to punctuality.
Strangely, a large hill happens to lie between my place and Aunt Sally’s house. This hill is fortunately regular in its shape – it ascends for two kilometres at a 45° incline, making it very difficult and slow to climb. However, the final two km are a 45° slope downward on a good path, allowing one to easily make up for any lost time on the ascent.
On this particular day, I climb the initial slope at a rate of 3km an hour – how fast will I have to descend in order to average 6km an hour and make it to Aunt Sally’s on time?
‘Mkay….what number do you have in mind?
If the answer is “any”, then you are wrong. I could descend that hill at the speed of light and still be late. A fraction of a second late, granted, but late enough to incur the wrath of Aunt Sally.
The correct answer is that I have already failed, any speed less than ∞km an hour will fail to maintain that average. Or, to put it another way, if I need to cover any distance at that rate of 6km an hour, and I cover half of that distance at half of my goal speed, then I have already used up all of my allotted time.
This trick question relies on very different techniques from the Moses question posed above. Here, the information is obfuscated with additional, irrelevant information (the incline of the hill’s slopes is unnecessary), which is further complicated by the fact that people are, generally, quite bad at maths. A common automatic response is to assume that going at 12km/hour is the right answer, since if the pace was halved before, then doubling it ought to ‘make up’ the difference. But this is a problem of addition, not multiplication, despite the fact that the method for determining km/h relies upon division.
In short, then, trick questions rely upon using assumption against the solver.
What does all this have to do with my little puzzle? Well, simply that the actual, relevant information is here encoded in the format of the puzzle. See, a third type of trick question is to use convention and form against the reader – which is how the present puzzle is presented. But in order to make sense of that, we need to figure out which statement in the puzzle is false – and, for that matter, how many statements there are in this puzzle at all.
It might look like there are 3 to choose from, but I would argue there is a 4th. After all, the statement “One of the statements in this puzzle is false, while the others are true” is itself a statement, and thus is itself included in the puzzle.
I tried to include a couple of clues to this in the other statements, as well – the first statement is self-demonstrating, as is the fourth (though we’ll come back to that) while the second includes the number “4”.
Now, all of this may have been obvious to you, and fair play if so – I don’t think that it was the most carefully or cunningly hidden trick. But, in a way, the trick question is itself a trick…or at least, it’s a ruse, intended to lead one into looking for more tricks.
But first, let’s break down the consequences of the 4th statement being part of the puzzle.
If statement 4 is true, then one of statements 1-3 is a lie, while the other two statements are true, as is statement 4.
Statement 1 is true. So is statement 2, and also statement 3. This means that statement 4 is not true, and we can get on with our lives.
Except…if statement 4 is untrue, then statement 4 fulfils its own claim. 3 of the statements are indeed true, statements 1-3, while one statement is false.
Therefore, statement 4 is true, because it is false. But if statement 4 is false, then it cannot be true, which means it is false, which means that all the statements must be true…
In short, this is a modified form of the Liar’s Paradox – so that means it’s time to talk paradoxes.
I love solving trick questions…but, equally, I love not solving paradoxes. Or, rather, questioning and attacking their logic, and seeing what strange results come out. This particular puzzle is a homage to one of the best-known and most often misquoted paradoxes – the Liar’s Paradox.
The Liar’s Paradox is often presented in this format:
“Everything I say is a lie”
This is a terrible way of presenting it, as it introduces ambiguity. If everything you say is a lie, then you are telling the truth…which is not a lie, so you could not have told it. But, if only some things you say are lies, then you can have lied about saying that everything you say is a lie. If you have uttered at least one truth before, you’re in the clear and the claim is not paradoxical.
No, the original presentation of the Liar’s Paradox is less catchy, but also much more rigorous:
“This statement is a lie.”
If that is true, then the statement is a lie, which means it is not true, making the statement true…
Mind, I wasn’t thinking about any of this when I conceived of my puzzle – I just wanted to try and make a tricksy little brainteaser with a paradox at the end of it. It was only when I sat down to write it out that I realised that its construction is identical to the Liar’s Paradox.
Nonetheless, I’m quite fond of it as a method for presenting the Liar’s Paradox – is it bad faith to present a puzzle that is technically unsolvable? Perhaps – but I liked the idea of it, and I enjoyed formulating it enough that I wanted to share it somewhere.
So, there you have it – my version of the Liar’s Paradox, nestled inside a trick question, nestled inside a regular puzzle. I think it is rigorously constructed, but I am not a puzzle-maker….in any case, my apologies for any errors of logic in it, and I hope the puzzle (and perhaps even the explanation) have been mildly enjoyable for you!
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