You are desperate to get past the guard and get into the party at The Sink (You're you, so you know why). To get in, you need to give the right password to the guard. You watch from the side at all of the guests coming in.

The first guest walks up to the guard. "Eleven," says the guard. "Four," responds the guest. He is allowed in.

The next guest comes up. "Twenty-nine." "Four." He walks in.

Next guest. "Nineteen." "Three," replies the guest. Made it in.

Next guest. "Fifty-six," says the guard. "Two," she replies. She is asked to leave. She must have been trying to figure it out too.

Next guest. "One hundred seventy-seven." Woah, big one. "Three," replies the guest. She is let in.

Another guest. "Four." "Zero," the guest replies. He is let in.

You start to recognize what may be going on. You get up and approach the guard. "Three hundred ninety-one thousand, eighteen," he says. Jerk.

(Admittedly not an original premise, but a good one)

More Correct Responses:

Two million ~ Four
Eight thousand ~ Four
Thirty-seven ~ Three
One ~ Three
Five ~ One

Hint 1:

This place's passcode will likely be very different than their rival business The Source...

Hint 2:

It seems some of the guests were counting something on each hand...

• @Asterisk Numerical form is 391,018 Jun 17, 2019 at 14:15
• Are the sample correct answers in the hint only valid in sequence, or are they standalone? Jun 18, 2019 at 10:53
• @Mohirl Standalone, not in sequence. Jun 18, 2019 at 15:01

six

Count the number of letters in the word when written out. then count the number of letters in that word when written out. Continue until you get to 4, which has 4 letters. The number of numbers you visit, other than 4, is the answer.
391018 -> 37 -> 11 -> 6 -> 3 -> 5 -> 4
has six steps.

Alrighty, so I actually had stumbled upon the right answer, as @Nordii mentioned after correcting the question.

6

This is because:

You apply the pattern "Count the number of letters in the number, then apply it".
For example: "eleven" would be "six" because "eleven" has 6 letters.

Then you continue applying that pattern until you reach the number "four" because that will result in an infinite loop ("four" has 4 letters). The answer is the number of times until you reach four (the "sink" as it's an infinite loop).
For example: "eleven" -> "six" -> "three" -> "five" -> "four". Given there were 4 "transitional" periods, this gives you the answer 4.

"Three hundred ninety-one thousand eighteen" -> "thirty-seven" -> "eleven" -> "six" -> "three" -> "five" -> "four"
As there were 6 transitions before arriving at "four", the answer is 6.

Starting Thoughts

Given Hint 1 and the title of the question:

This problem could have to do with either a Computer-Science idea of "flow" (describes stuff "flowing" from a "source" to a "sink"), electronic flow, or thermal (heat) flow, or any idea that has to do with something flowing from a "source" to a "sink".

Given Hint 2, and Mohirl's observation that "All the responses are..."

All the responses are 0 to 4 (Thanks Mohirl!), which is something one can count to with their hands

This gave me a few ideas, which unfortunately didn't pay off but could be useful still as something to spark ideas:

- Number of syllables in the words, showing how the flow of the mouth goes. Immediately broken by "eleven".
- Number of times your tongue touches the roof of your mouth when you say the number. Also immediately broken by "eleven".
- Comparison of consonants vs vowels. Also broken immediately by "eleven".
- The classic "count the number of letters in the number, this gets you the next number". Then you see how many times you apply this until you get to 0. So "eleven" would be "six" because "eleven" has 6 letters. So it'll be "eleven" -> "six" -> "three" -> "five" -> "four". Given there were 4 "transitional" periods, this gives you the answer 4. This looked promising until I tried "twenty-nine", and got "four" instead of "three". ("twenty-nine" -> "ten" -> "three" -> "five" -> "four"). EDIT: This was actually a mistake in the question, the actual answer is 4.

Notable clues:

Hint 2 strongly suggests it can be simplified by doing something with your hands, so they must be counting something with those numbers.
"Four" -> "zero" is incredibly interesting. This is the only time it was answered with "zero", so what's so special about four? "one" -> "three" suggests it has less to do about the numerical value, unless they add/subtract something.

EDIT: Adjusted my answer thanks to Nordii's correction that "twenty-nine" -> "four".

• You caught an error in my question. Twenty-nine ~ Four is correct. Good find. Jun 21, 2019 at 14:52
• @Nordii Oh, lol. I wasn't sure if I was on the right path. I'll try my answer to reflect that.
– Jay
Jun 21, 2019 at 21:01

Partial / Starting point thoughts

As a starting point, given hint 1

and the name The Sink, I'm guessing the response is subtracted from something related to the question, to give a target answer

I'd initially thought that the

He vs She in the story examples might be relevant, but since the "More correct responses" doesn't include that, it's clearly not

All the responses are

0-4, so something mod 5 might be involved?

There's no

clear pattern to the sequence of question numbers, in terms of increasing/decreasing, odd/even, number or sum of digits