Zed's Phone: Failure to Communicate

Question

My friend Beto and I constantly buy things for one another. One day, he gave me a strange-looking phone and said it was my birthday gift (My birthday wouldn't be until 3 more months). He said he bought it from some weird store and it was made from some company called Jasoom; I, of course, never believed him, but I kept the gift anyways as a token of his appreciation.

Years later, while rummaging through some of my old things, I happened to find this phone again! Reluctantly, I turned it on, finding out that I had received a single text message. It read:

69474727250258837936906947725583?

I stood and stared for a moment until I discovered what the other person was asking, only to realise that the phone I had was using a different set of keys --- apparently whoever manufactured my phone installed the keypad backwards! I fumbled with the controls until I produced a half-hearted response, which said:

749097359606274852852

I was a bit worried at first if I mistranslated their message (or if they would even bother with a response well after they sent their original message), but much to my chagrin the person on the other end responded in less than 10 seconds flat! I knew for sure they understood me, because this was their response:

472369694706947369369,07254722583690583725072536936958306947725583

Months went by until they texted me again, although this time with a far-more alarming question, which read:

694747225858308379694258725369583025883793690694772558303698379725725?

I was completely flabbergasted, so I offered the only response that made the most sense:

8529733852749749

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Quite the "close encounter", wouldn't you say? Anyways, while texting my compatriot, I realised something I never noticed before:

One of the labels on the keypad was scratched off (didn't I say this phone was poorly made?), so I could not read what it originally said. Fortunately, I never used the key during my friendly discussion, but I would still like to know what was that missing label... perhaps you could help?

You only need to give me the sequence that was used for the label, not the corresponding letters; however, I'll show preference to any answer that explains which characters were used as well.

---

Edit: Turns out I made a mistake in my original, hand-written notes and accidentally mixed up one of the key sequences when encrypting; it's fixed now, but I wanted to inform all of you who may still be working on the problem before you get led too far astray.

I apologize for any confusion.

---

HINTS:

While it may not be immediately obvious, when I wrote 6274852852 and my compatriot wrote 6947369369, we were saying the same thing; however, when they wrote 6947472725, nothing matched what I wrote.

Just to be clear, we were both talking about me... and I kinda lied (something matches, but not the whole).

Funny thing I realized, though, following my conversation: if, perhaps, this phone was manufactured by a different company named Barsoom, my last response should probably have been 59697338523852... and – even more surprising – I still would not have used the unlabeled key!

I think it's necessary to remind you that this is a phone; as such, keypads should have labels of only one of two different lengths (or three, if you're being a smart-alec about characters). Of course, I'm not telling you how long the missing sequence is, but perhaps that narrows down the field considerably.

Still not convinced? Fine, think about how a normal phone-pad is arranged, then think about how it's labeled; you should be able to find the numbers that are missing this way, but this may not be enough to tell you how they are ordered or which key indicates which sequence.

Answer 1

Solution:

Which label had been obliterated on Zed's phone keypad?

The key labeled 2 with the letters TUV.

This is a guess. See below.

Actual deduction:

The key labeled 638 with the letters JKL.

Plaintext:

Beto: WHO ARE YOU?
Zed: I AM ZED
Beto: HEY ZED, NICE TO MEET YOU
Beto: WHAT PLANET ARE YOU FROM?
Zed: EARTH

plain   beto  zed
 ABC    258   973
 DEF    369   852
 GHI    472   749
 JKL    694
 MNO    725   596
 PQRS   8379  3852
 TUV    583
 WXYZ   6947  6274

Solve path:

Setting aside most of the story and focusing on the digits, I first searched for repetitions and found the text was extremely repetitive.
I noticed that the repetitions tended to align in three-digit groups, and that after highlighting these groups the zero digit seemed to act like a word separator. This also revealed that Beto's messages and Zed's messages used a completely different set of three- and four-digit groups:
With no more than ten unique groups per person, I assumed each group was both a substitute for a letter and a placeholder for a single digit. I assumed the first message read WHO ARE YOU? and the second began with I AM. I remembered that the story talked about phone keypads, and noticed that the letters assigned to the four-digit groups looked like pqRs and WxYz. Assuming that each group substituted any one of three or four consecutive letters led to a nearly-complete decipherment.

I noticed that Beto's "code-groups" usually began with the keypad digit that is normally labeled with that letter, and is followed by the two digits directly below that key, wrapping around to the top of the column on the right. With two exceptions: JKL is substituted with 694 where this pattern expects the group 583, which Beto uses for TUV. Similarly, WXYZ is 6947 when it ought to be 9147 or possibly 9472.

While assuming Zed's last message read EARTH (or MARS following the second hint), I found I had to assign both T and H to the group 749, which violated this pattern, as does Zed's 973 for ABC. Now, I had to consider that the letters are in the expected places on the keypad, but the digits are out of place, leading to different sequences down the columns. Or perhaps the keys are not on a square grid at all, which leads to both 973 and 749 being sequences down the "columns". I'm not sure I have enough information to deduce the missing key label here, so I have to guess that the answer is the key is missing the label TUV and the digit 2, keeping in mind that you

kinda lied

... when you said that you never used the key, because you did use it to continue the code group 852 for DEF, and you would have used it to write the letter T in EARTH if you knew how it was labeled.

Update:

After writing out all of Beto's code groups below the normal alphabet, and writing out Zed's code groups below a reversed alphabet, a pattern emerges. Shifting Zed's alphabets to the right by six places will align all of the digits except for two, which have swapped places. If I assume this is a mistake in the puzzle, I can deduce that the two apparent unused keys on Zed's phone are JKL=638 and TUV=749. This causes GHI and TUV to use the same code group, which clears up the confusion when the word EARTH was assumed, and leaves JKL=638 as the single, unused, obliterated key.

 Beto    GHIJKLMNOPQRSTUVWXYZABCDEF
 Beto    47269472583795836947258369
                    !            !
 Zed     4726   2583695   947258379
 Zed     ZYXWVUTSRQPONMLKJIHGFEDCBA
 

Alternately, if the single digit is not a mistake, I can wildly rearrange Beto's alphabet to match Zed's exactly, and still get the exact same result with JKL=638 and TUV=749

Beto    GHIWXYZABCDEFTUVJKLMNOPQRS
Beto    47269472583695836947258379

Zed     4726   2583695   947258379
Zed     ZYXWVUTSRQPONMLKJIHGFEDCBA