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“We get a lot of applicants for our job vacancies,” said Lionel. He waved a hand in the general direction of the call-centre. “Here at Open to Everybody Insurance Services the jobs don’t need much prior knowledge and we offer some great training and near-benefits: for instance, almost all of our staff have a chair to sit on this year. But it creates a problem for HR when they’re trying to choose candidates.”

“It sounds like you just need to do a quick check that they’ve not got any major socialising problems,” said the consultant. “Surely that’s not why you’ve hired me?”

“HR don’t like talking to people,” said Lionel. “For a long time they used to just shake all the CVs we received over the bin. The ones that fell in were the ‘unlucky’ candidates and were rejected. The rest were accepted.”

“That sounds—”

“Efficient? Yes. But then we got audited and told we had to be more…”

“Humane?”

“Scientific. So now HR has a simple incremental policy: each candidate is numbered in increasing, sequential order. They accepted candidate number 1 and rejected candidate number 2. Then they accepted candidates 3 and 4 and rejected 5 and 6, and so on, always increasing the accept and reject quantity by 1 each time. They’ve just rejected candidate 27.”

“Right….” The consultant checked his watch, wondering when he could leave.

“The problems are these: first, I know that our rival, Wythoff, have just laid off a number of people. I don’t want to hire any of them as they’ll all be spies. There are 10 of them in total, and I need to know what numbers to give them to get them all rejected as a group so it doesn’t look suspicious, and as soon as possible.”

“So you want the start number for the first contiguous block of 10 rejects?”

“That sounds fancy, so yes. I can adjust the numbering in the HR system without getting caught so long as it’s below 250. Second, I want to give my friends bonuses when I hire them. I need a way of designating certain numbers as ‘lucky’ and then I’ll assign bonuses to the lucky numbers. It’ll look arbitrary to HR, but I’ll assign those numbers to my friends. And the numbers should get further apart over time, so it’s harder to spot a pattern.”

“When you say ‘certain numbers’, can’t you just say ‘the first number in an accept block?’”

“No. HR will see right through that; they’re lazy, not stupid. I need something like ‘every other prime number’, or ‘numbers congruent to 1 modulo my age’.”

“How old are you?”

Lionel gave the consultant a look that suggested they would be waiting a very long time for the answer. “Just get on with it.”

For the tick: what's the start number of the first contiguous block of 10 rejects, and what is a simple formula that will only produce numbers that are accepted and are spaced increasingly further apart?

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First block of 10:

$101$

A simple formula for only accepts:

$n^2$

How:

Look at the list of first rejects, it is $2,5,10,17,\dots$, which is $n^2+1$. So $n^2$ is always accepted, and $10^2+1=101$.

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  • $\begingroup$ Very nice, and the logic is sound! $\endgroup$ – postmortes Sep 23 '19 at 15:32

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