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Grandpa e mailed me a puzzle. Seemed very straightforward to me until I read it again.

"Are you sure this is right?" My e mail

"If there is a mistake- and I'm not saying there is- then it might lead you to the answer!"

This is what he sent me.

21 begins with a T

311 begins with a T

511 begins with an F

31 begins with an S

51 begins with ? Is it E or I or O ?

What do you think? What is the logic?

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    $\begingroup$ No 3D or Phone Codes. $\endgroup$
    – DrD
    Commented Jan 25, 2021 at 23:08
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    $\begingroup$ Are these indices? The obvious answer is that if "511" begins with F so must "51" but that isn't an option. $\endgroup$
    – humn
    Commented Jan 25, 2021 at 23:16
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    $\begingroup$ There is a fun clue in the puzzle. It was my "aha" moment! $\endgroup$
    – DrD
    Commented Jan 25, 2021 at 23:23
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    $\begingroup$ That the statement separates "e" from "mailed" sticks in my craw. Everything else tickles me. $\endgroup$
    – humn
    Commented Jan 25, 2021 at 23:27
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    $\begingroup$ I was thinking about digit sums, am I on the right track @DrD $\endgroup$
    – Stevo
    Commented Jan 26, 2021 at 1:48

2 Answers 2

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I think the answer might be

O

Reasoning

Consider that, on a keyboard, the 1 is underneath the !
What if, instead of writing numbers like 3!, Grandpa forgets to use the shift button and writes 31 instead.
In this way, his numbers should read as follows

21 should be 2! = 2, which begins with T.
311 should be 3!! = 3, (using double factorial) which begins with T.
511 should be 5!! = 15, which begins with F.
31 should be 3! = 6, which begins with S.

and so

51 should be 5! = 120, which begins with O.

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    $\begingroup$ Ah, this has to be it, surely! I was getting so consumed by the sequence I'd identified in comments above for the RHS that I couldn't see anything else - nice spot :) $\endgroup$
    – Stiv
    Commented Jan 26, 2021 at 12:51
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    $\begingroup$ Xabjvat Tenaqcn, ur jbhyq sbetrg gur fuvsg ohggba vagragvbanyyl!! be vagragvbanyyl11 $\endgroup$
    – DrD
    Commented Jan 26, 2021 at 13:29
  • $\begingroup$ That's some obscure math notation. In other contexts it would actually be read as a different operation. $\endgroup$
    – Joshua
    Commented Jan 26, 2021 at 21:17
  • $\begingroup$ @DrD Ohg qvqa'g Tenaqcn hfr na rkpynzngvba znex va uvf ercyl gb lbh? $\endgroup$
    – user
    Commented Jan 26, 2021 at 21:56
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    $\begingroup$ Nf V fcrphyngrq orsber Tenaqcn zhfg unir qbar gubfr guvatf gb vagragvbanyyl qvfgenpg naq gevpx zr. @user $\endgroup$
    – DrD
    Commented Jan 26, 2021 at 23:06
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It seems too straightforward but here is my idea:

Twenty one : T
Three hundred eleven : T
Five Hundred eleven : F
Thirty one : T, but Grandpa made a mistake and took the letter just before T: S
Fifty one : F, but if we make the same mistake we get E

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    $\begingroup$ Or "S" for "Sorry, I can't remember how to spell 31" which leads us to "Oh well, same for 51, damn Alzheimer's disease"... Yes a bit too straightforward but may interest others anyway. Welcome to puzzling.SE! $\endgroup$
    – xhienne
    Commented Jan 26, 2021 at 10:14
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    $\begingroup$ To whoever upvoted: Thank you but you didn't have to, I knew it was too literal and most likely wrong when I posted this answer (maybe I should have specified "initial idea"), so I don't mind it being downvoted :) It will take more than a few downvotes to scare me away! $\endgroup$
    – Pepper
    Commented Jan 26, 2021 at 11:15
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    $\begingroup$ That was a welcome vote, it only happens once in a lifetime :-) Your reasoning is not that bad; I guess it is the same reasoning that led the poster ask "Are you sure this is right?" Now try to figure out Grandpa's answer... $\endgroup$
    – xhienne
    Commented Jan 26, 2021 at 11:22
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    $\begingroup$ Welcome to PSE, Pepper :) Some inside information: DrD's "Grandpa" puzzles (long-established here) usually involve some clever trick of lateral thinking. There will turn out to be something where this sequence of statements makes perfect sense, and he has deliberately chosen cases which appear to satisfy a different rule on the surface (namely initial letters of numbers)... until a certain point when your mind goes "What?!" Your answer here if correct would definitely be far too straightforward for one of these puzzles (as you suspected). But keep thinking! $\endgroup$
    – Stiv
    Commented Jan 26, 2021 at 12:02
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    $\begingroup$ @Stiv Thank you for the context, I'll be eagerly waiting for Grandpa's next puzzle! (and taking a look at the previous ones too) $\endgroup$
    – Pepper
    Commented Jan 26, 2021 at 13:49

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