The Doppelganger

In the early 1910s, when aristocrats from European countries were murdered every few weeks, authorities began to fear the roots behind an ancient German myth. The murders themselves did not scare anyone, given the constant conflicts experienced in these countries, however, the nature of these crimes was strange to explain, since a person close to the victim, his family, friends, acquaintances, or in his service, he suddenly committed this crime. Although betrayals were not new to history either, the strange thing about it is that the alleged traitor was found dead longer than the crime itself occurred.

It was clear from the victims which families benefited from these crimes and who were probably the masterminds. Insider information indicated that the principals did not know the killer, only his intermediaries. However, the authorities, though normally skeptical, have begun to believe that the killer might have some supernatural power. Something that allows her to change her appearance and alter her face, becoming a lookalike to someone close enough to the victim to murder her, associating it with the Doppelgänger myth itself, which heralds an omen of death when meeting someone identical to someone else. Despite the effort to refute the theory, the pattern of these murders continued to be repeated in various parts of Europe, with people with very different profiles (men, women, elderly, fat, thin, tall, short…). Even illusionists of the time agreed that it was absurd to think that a mere disguise could change someone’s appearance so much.

Investigations into the Doppelgänger continued, authorities even sought reinforcements in the medical departments of several renowned universities in order to understand the nature behind how this killer altered his appearance. However, at one of these universities, the departments of medicine and mathematics had a good dialogue, so the story reached the ears of some mathematicians who immediately suggested that the initial hypothesis might be false. Authorities almost rejected the group that apparently doubted the seriousness of the police work. However, mathematicians explained themselves by saying that if the hypothesis that there is a single killer is true, they would need to show that there is at least one human being with Doppelgänger’s ability. But if the hypothesis is false, we need to explain how an intermediary found for each aristocrat a murderer resembling someone close to him.

Although short, this conversation brought to the authorities another way of looking at the problem. Resuming their investigations, they realized that they were missing the obvious, and doing fieldwork found that all the victims were visited by representatives of a photography company that advertised their revolutionary devices. On this visit, they demonstrated their equipment by taking pictures of everyone in their family, friends and employees. The intermediary had access to a large network of assassins, and was looking to match between the photos of available assassins and those close to the victim, some very similar pair. Making it easier for the killer to reach the victim and dispersing the attention of investigations based on the hypothesis of a killer capable of changing his appearance.

About the post

This is a fictional tale, but it discusses some very interesting aspects of mathematics and research.Well substantiate a hypothesis. When the authorities came across the reports of the crimes, they readily assumed that it was a single murderer, thus adhering to the hypothesis of a Doppelgänger. I say this because sometimes we start with very shallow aspects, common sense or based on our personal beliefs, and from them we base hypotheses “that we like”, but they are not really good hypotheses.

1. Well substantiate a hypothesis. When the authorities came across the reports of the crimes, they readily assumed that it was a single murderer, thus adhering to the hypothesis of a Doppelgänger. I say this because sometimes we start from very shallow aspects, common sense or based on our personal beliefs, and from them we base hypotheses “that we like”, but they are not really good hypotheses.

2. Identify influencing factors. In this tale, the authorities ignored the events of the victims’ families, considering the crime disassociated with what took place before the incident. In this case, there was a directly related factor that was hastily ignored (people close to the victim were identified in detail through a photograph). This factor that initially seems unrelated, when considered, could point out why some were victims and others were not.

3. Extra-peer communication. Although discussing our investigations with peers is simpler, after all, they are already used to that repertoire of concepts, when we take the case to extra-peers, we have a completely different perception of the subject. Sometimes it seems a hostile or petty perception, but this has to do with the very nature with which each field of knowledge analyzes the topic. In the story, the authorities had the fixed idea that there was a Doppelgänger, and they were trying hard to prove it. They looked for illusionists to ensure that one person could not disguise himself as so many others. They sought out biologists to understand the physiology of a human being capable of altering their appearance. But it was a first look by mathematicians who maintain extra-peer communication with biologists that suggested the initial hypothesis was false.

4. Ockham’s Razor. A principle of scientific research that dictates the choice among several explanations for a phenomenon, one that depends on the least possible number of variables and hypotheses. In the case of the short story, the simplest explanation was that a single murderer disguised himself to carry out the crimes, but since it was ruled out by consulting the illusionists, the authorities remained stuck in the chance of a single murderer, but inserted variables of that he could alter his appearance through some supernatural ability. This is in fact a hypothesis that depends on a series of other more complex variables, and is thus “pruned” by Ockham’s Razor. Recognizing that we need an explanation for this scenario, the simplest thing in this case is to rule out the hypothesis of a single murderer. Although it still requires explanations on how to associate them with people close to the victims, this is certainly more natural to be explained than the existence of Doppelgänger’s ability.

Now a little about math

The idea behind this tale involving murderers looking like people close to the victim is based on a similar concept to that of the birthdays paradigm. If we consider the appearance as “birthdays”, when comparing its appearance with that of another set of assassins, there is a chance of matching related to the number of assassins. That is, thinking that the victim is Earl John, the chance of comparing a random person with the appearance of your wife and finding a murderer who looks close to her is x%.

Thus, the chance of comparing a random person with the appearance of the Earl’s wife and NOT finding it, is 1 – x%.

But when we consider a list of N random-looking assassins, the chance of not finding a person who looks close to the Count’s wife is (1 – x%)^N.

If this x% is for example 0.01%, and the number of assassins is 500. We have to (1 – 0.01%)⁵⁰⁰ ~ 95%. In other words, we would have a 5% chance of finding someone who looks close to the Earl’s wife.

Now, if we think that we have M people close to the victim, the chance that none of the N killers look like any of the M people close to the victim is given by:

[(1 – x1%)^N][(1 – x2%)^N][(1 – x3%)^N]…[(1 – xM%)^N], where x1, x2 , …, xM is the chance that each person next to the victim looks similar to another random person.

If we assume that this x% is 0.01% for all, and that we have 100 people close to the victim, then we have ((1 – 0.01%)⁵⁰⁰)¹⁰⁰ ~ 0.6%. That is, a 99.4% chance of finding a killer who looks like someone close to the victim.

Cover image credit to Jenő Szabó from Pixabay