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.
