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"As per corporate policy we need to change the passcode for the building's entrance every six months," said Lionel. "Your job is to pick the new passcode. It must have at least four numbers in it."

You smile; this is definitely the easiest task you've been given to do since joining Coldport Research and Development. Your smile fades slightly as Lionel continues talking however.

"Ms. Salem is the deputy head of IT," he says. "She must approve the passcode and she has absolute power of veto. Mr. Spencer is the head of IT. He must also approve the passcode and also has absolute power of veto."

Your confusion must show on your face as Lionel holds both of his hands up, palms towards you. "Because of the Incident," he says. "Which we don't talk about."

That makes fourteen Incidents-we-don't-talk-about so far, but you keep quiet. "This is the panel we enter passcodes on," he says. "Type the code and press the Red ENTer key."

enter image description here

"The last two approved passcodes were 1,2,4,5,10,11 and 1,2,7,8,10,11," he says. "Ms. Salem rejected 1,2,4,5,10 when it was proposed, as well as 1,2,4,8 and 3,4,6,10,11. Mr. Spencer rejected 1,2,3,7,8,11 and 1,2,6,7,10,11. Both of them rejected 1,2,3,4,5 which, to be honest, I was quite glad about."

Find a passcode that both Ms. Salem and Mr. Spencer will accept, and indicate why each of them will find it acceptable.

Hint 1:

Ms. Salem's office, you notice, is cluttered: if she has only one of something it's because it's unique, whereas Mr. Spencer's office is chaotic: there seems to be neither rhyme nor reason to where things are or why they're there.

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  • $\begingroup$ Would Mr. Spencer have accepted all of the sequences that Ms. Salem rejected, and vice versa? (ie. Mr. Spencer would be a fan of 1,2,4,5,10; 1,2,4,8; and 3,4,6,10,11; and Ms. Salem would have been happy with 1,2,3,7,8,11 and 1,2,6,7,10,11?) $\endgroup$ – El-Guest May 12 '20 at 17:13
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    $\begingroup$ @El-Guest yes. The ones they both rejected (and accepted) are explicitly mentioned. $\endgroup$ – user40528 May 12 '20 at 17:18
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Ms. Salem

Seems to only accept passcodes that are 6 digits long. Since she’s a clutter bug, she likes lots of numbers.

Mr. Spencer

Seems to reject any passcode where three of the digits form an arithmetic sequence. Since he’s chaotic, he doesn’t like ordered sequences.

Therefore a passcode that should work is

1,2,4,8,10,11?

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  • $\begingroup$ You're getting much, much closer; Mr. Spencer and Ms. Salem are unanimous in rejecting your code though, each for their own reason. (You are definitely along the right lines with Mr. Spencer but you're being perhaps a little too specific) $\endgroup$ – user40528 May 12 '20 at 19:04
  • $\begingroup$ How does this look, @postmortes ? $\endgroup$ – El-Guest May 12 '20 at 19:18
  • $\begingroup$ Congratulations! Mr. Spencer and Ms. Salem both nod approvingly, and the company has their new passcode! Hint 2, had it been needed, was to look google Spencer and Salem as the passcode has to a be a Spencer-Salem set -- a maximal set from the numbers $1-n$ (here $n=11$) that avoids all arithmetic progressions. $\endgroup$ – user40528 May 12 '20 at 19:22
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    $\begingroup$ Yay! Learned something new today — neat concept, for sure! $\endgroup$ – El-Guest May 12 '20 at 19:26
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I suggest

1,2,7,8 for example

That's because

1) Ms. Salem knows that due to some errors in panel schematic, the pairs of keys in 2 leftmost columns and the same row (1/2, 4/5, 7/8 and 10/11) should be used or unused entirely, so the rejects any combination which uses one of the key from a pair but not the other one (e.g. 4 but not 5, 8 but not 7 etc.).
2) Mr. Spencer knows that the rightmost column is faulty (e.g. pressing any key on it triggers the ENTER key too), so its keys (3,6 and 9) should not be used at all.
3) Finally, because both of them are IT heads, they know that using a combination like 1,2,3,4,5 which can be cracked in 1 attempt is very, VERY insecure, so they both rejected it immediately.

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  • $\begingroup$ Mr. Spencer nods approval. Ms. Salem does not like your suggestion and vetoes it immediately. Your reasoning is interesting, but the answer is not related to the layout or the functionality of the panel. $\endgroup$ – user40528 May 12 '20 at 5:50
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    $\begingroup$ Your rule for Mr Spencer doesn't seem consistent with the fact that Ms Salem (rather than both) rejected 3,4,6,10,11. $\endgroup$ – Especially Lime May 12 '20 at 13:17
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Well, the most obvious explanation for Ms Salem is that she rejects

any passcode with five or fewer numbers.

There are probably several possibilities for Mr Spencer, but I can't find one that seems as natural as the above. Certainly it's consistent with everything we've been told that he insists that

the average of the numbers is a (nontrivial, i.e. not an integer) terminating decimal.

If the above is what was intended, one option satisfying their requirements is

1,2,3,4,5,6.

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  • $\begingroup$ You're very much along the right lines with Ms. Salem, and when you work out what Mr. Spencer is doing you'll be able to complete what she wants in a flash. But Mr. Spencer isn't averaging anything and he rejects your new passcode with a curt shake of his head. Ms. Salem, though, is happy with it. $\endgroup$ – user40528 May 12 '20 at 13:20
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I believe that both will accept

1,6,7,8,9,10.

I agree with Especially Lime that Ms Salem

rejects any passcode with 5 or fewer numbers

I am less confident about Mr Spencer, as this seems rather forced, but it looks like

he accepts passcodes where the difference between two adjacent digits is 2 or 5

Based on both of these preferences

There are a large variety of possible answers, as any 6-digit series with a gap of 2 or 5 somewhere within will work. So, 1,3,4,5,10,11 should also work, for example.

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  • $\begingroup$ Hi! Can you perhaps explain your answer more as to your choice? $\endgroup$ – Prince North Læraðr May 12 '20 at 18:09
  • $\begingroup$ Ms. Salem nods approval at your code, but Mr. Spencer is adamant: he will not accept it. Your second guess is also rejected -- and though Mr. Spencer doesn't tell you exactly why, he will tell you he's rejecting them for the same reason. $\endgroup$ – user40528 May 12 '20 at 18:19

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