The mathematical crimes of the Young Sherlock Holmes series
Warning this article contains spoilers about the new Amazon Prime series Young Sherlock.
I’ve read the whole Sherlock Holmes canon multiple times over. I love how Holmes uses analytical reasoning to unravel problems that look mysterious, but ultimately prove to have simple explanations. So I was excited when I saw Guy Ritchie’s Young Sherlock appear on Amazon Prime. My excitement was quickly tempered when I started watching, though.
A key part of the plot relies on mathematics. Holmes first meets his sidekick Moriarty (yes, he is working together with his future adversary) at the blackboard after a maths lecture at Oxford. Despite some mistakes in the dialogue, the maths on the blackboard is interesting enough. It is finding the solutions to the equation x5 + x4 + x3 + x2 + x + 1 = 0. As shown nicely in this video, the equation has five solutions.
In the maths many of us will have learned at school, we are taught that a positive times a positive makes a positive and that a negative times a negative also makes a positive. For example, 3 times 3 equals 9, but -3 times -3 also equals 9. Squaring a number (when you multiply a number by itself) should always give a positive result. The reverse operation – finding the number(s) you multiply together to give a positive number – is called taking the square root. The two square roots of 9 are 3 and -3, since when you square either of these numbers you get the answer 9.
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If we want to take the square root of -1, say, then we need to venture into the realm of imaginary numbers. Imaginary numbers are the square roots of negative numbers. Mathematicians defined the imaginary number i to be the square root of -1 (technically -1 has two square roots i and -i). The square roots of other negative numbers are multiples of i. The square roots of -9, for example are 3i and -3i. Some of the solutions from the equation on the blackboard involve imaginary numbers (this will turn out to be an important plot point).
Mathematical blunders
It’s plausible that the equation on the blackboard might appear in an early first year undergraduate tutorial. Something approaching a passable solution is given, but in excruciating detail (the sort of detail you wouldn’t use at school, let alone in a maths degree at Oxford). And there are mistakes in the maths.
Amazon Prime screenshot
Towards the end of the lecture, the professor sets the students homework to find all the solutions to the equation, even though they are already written on the board (although incorrectly). Despite this, the end of the scene sees Sherlock spending some time trying to think of the solutions before Moriarty comes up and shows him two of the five solutions (as if they were the only ones). Moriarty too writes these down incorrectly, but in a different way to the incorrectness already on the board.
As Moriarty writes down the complex solution (complex means the answer contains both real and imaginary numbers) he says “These solutions, they’re not real. They’re imaginary.” which we can allow (although technically he means complex).
What we can’t forgive is Moriarty going on to say, “That means even if you can’t see the target, you can still shoot for it.” Which is nonsense, even as a metaphor. Complex numbers aren’t targets you can’t see, but well-defined, mainstream (even in the 1870s) mathematical quantities and there’s no sense in which you “aim at” a complex solution to an equation.
Death by numbers
In the last episode, Holmes and his team are battling to halt the distribution of a deadly chemical weapon known as the “creeping death”. They find a scrap of paper in a secret room which they say is the “equation for creating the creeping death.”
I was expecting to see some complex chemical reaction formulae sketched on the page, but when it’s held up to the camera, we see instead a mathematical equation: z3 + 4 z2 – 10 z + 12 = 0.
What does this have to do with the chemical process for creating the deadly nerve agent?
Nothing, it turns out. Or at least nothing I can imagine. In fact it’s a device to allow Holmes and Moriarty to hark back to that moment in the lecture theatre when they first met. What follows goes beyond artistic license into the realm of gibberish.
“If we have the positive equation”, they say, “then we can come up with the negative. And thus create a compound to neutralise the threat of creeping death.” Perhaps they meant “positive solution”, because equations themselves aren’t positive or negative. Either way, the idea that this simple mathematical equation or its solutions are the secret formula for making a weapon of mass destruction doesn’t make sense. There’s no context, no sense in which this equation could be the secret recipe for creating the nerve agent.
Moriarty points out that they have a problem. “This equation is not finished.” By this I think he means that the three solutions to the equation are not written out explicitly.
One solution, z = – 6 is given. And it’s correct. The rest of the scrap of paper contains a reformulation of the equation (a factorisation), which shows that the remaining solutions can be found by solving a quadratic equation: z2 – 2 z + 2 = 0.
A quadratic equation is just an equation built around a squared term (in this case z2), which has two solutions. The formula for the solutions may be familiar to GCSE students (normally aged 15 to 17 years old). For a general quadratic equation: a z2 + b z + c = 0, the two solutions are given below.
Yet, we are supposed to believe that, despite having supposedly solved a far more complicated equation than this in the first episode, Moriarty can’t find the solution to this much simpler equation. So stumped is Moriarty – the future maths professor – that he spends precious time, as a bomb is about to detonate, searching for a piece of paper with this missing solution. He almost loses his life when he could have just used a GCSE-level formula.
The piece of paper he eventually finds contains an incorrect statement of the quadratic formula alongside some nonsensical text, although the solutions are at least correct: z = 1 + i and z = 1 – i (where i, remember, is the imaginary number).
I appreciate my dissection of the maths is high-grade nerdery. Most people will have watched the series without pausing it like I did to look at the maths and probably won’t have noticed. But, if maths is going to be a pivotal plot point in your blockbuster series, then you’ll probably want to make sure you get it right.