As Numberland’s best detective, you thought you’d seen it all.
But the desiccated corpses of prominent natural numbers—
whole numbers greater than 0—
have been showing up all over the city for the past week.
All of Digitopolis is on lockdown from sundown to sunrise,
and it’s still not enough to stop what can only be described
as a vampiric feeding frenzy.
The police are helpless, and have turned to you.
You put the missing persons case you’ve been investigating on hold
and study the evidence.
Here’s what you know:
most victims have been completely drained of their life’s essence.
Some have a thin line of slime separating digits—
some of which were drained and others spared.
Which leads you to mystery one:
what’s the significance of the slime-lines,
and why are certain digits untouched?
Pause here to figure it out yourself. Answer in 3
Answer in 2
Answer in 1
Let’s start by looking for a pattern in the victims.
Why have some been fully drained, and some slimed?
Well, the slime-lined are in their prime—
which is to say they’re prime numbers, divisible only by 1 and themselves.
Not a single one of the unslimed numbers, on the other hand, is.
Furthermore, the drained parts of the slimed primes are not, themselves, prime.
When 71 got slimed, 7 was prime, but 1 was not.
If 461 had been split into 4 and 61, the right side would have been prime,
but instead it was split further into non-prime digits.
And when 2099 got slimed, the attacker went with 20 and 99,
and was able to drain both.
So you have a big problem.
There are infinite potential victims:
all non-prime numbers,
and all prime numbers that can be slime-lined
to reveal at least one non-prime number.
In fact, you can only think of a few numbers that aren't vulnerable.
First of all, the one digit primes: 2, 3, 5, and 7.
Then primes like 23, which could only be split into 2 and 3, both prime.
These numbers are familiar, but why?
Aha! Your missing person’s case.
Someone is kidnapping the numbers that can’t be drained:
we’ll call them the immune.
If you can identify the four immune who remain,
perhaps you can get to them first and figure out what’s going on.
Who are the immune? Pause here to figure it out yourself.
Answer in 2
Answer in 1
There are 4 one-digit immune: 2, 3, 5, and 7.
All have already been kidnapped.
But every digit in all of the immune must be one of those four numbers;
otherwise there would be a non-prime split possible.
That means there are only these 16 candidates among the two-digit numbers.
The ones that are prime must be immune.
We can eliminate anything that ends in 2 or 5,
as they’ll always be divisible by 2 and 5.
This diagonal is also out, as they’re divisible by 11.
If the digits of a number add up to a multiple of 3,
it’s divisible by 3.
So that eliminates 57 and 27, leaving these numbers.
Which, upon double-checking, are, in fact, immune.
We now have a strong starting point for searching three-digit numbers:
any 2 consecutive digits must be on the list of the two-digit immune.
Those all end in 3 and 7,
so the only possibilities are these 8 options:
the two-digit immune with additional 3s or 7s on their ends.
3 can’t follow 3, nor 7 follow 7.
And once again, adding the digits reveals that 237 and 537
are divisible by 3.
737 is 67 times 11, so that leaves just 373,
which is, in fact, prime, and therefore immune.
Could a four-digit number be immune?
No, any consecutive three digits would have to be immune,
and there’s no immune three digit number that starts with a 7.
You rush out, but reach each home too late,
until the only immune left is 373.
You’re in the nick of time to scare off her would-be kidnapper;
not the vampire itself but a minion.
373 explains that she and the other immune are the subjects of an ancient prophecy
that designated them as vampire slayers.
They’d sometimes gather and try to figure out what a vampire is,
but no one had the slightest idea.
They found the idea of slaying things quite distasteful.
You both trail her would-be kidnapper back to a foreboding castle.
There you release the other kidnapped numbers
and confront the vampire himself.
He’s powerless before the slayers, but ever the pacifists,
they round him up, return the lifeforce to his victims, and send him packing.
Numberland is safe... for now.