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You pick up a discarded newspaper, hoping to satiate your need for puzzles with a cryptic crossword. But as you turn to the page, you feel something is awry – perhaps some elision?

Cryptic Crossword?

Across
3. $F_6,F_7,F_8$ mixed $13\times3$ down${}- \sqrt{12\text{ across}}$
16. $3$ squares joined${}- 4$ and half of $3$ down
12. A square $6$ under $1$ across
20a. Multiple of $268727$ less than a square
8. Double $37+1019$a across - a square
19a. Code for a Lao $45$b down, head rooted
21. Twice $120$a across’ smaller prime factor
1. $17$ down${}+ 4163^\text{rd}$ prime
18. A square two-thirds of $5$d across
5d. Palindrome two-thirds of $1$ across${}+ 2$
19b. Multiple of $717$ down${}+ 10 $

Down
3. $\sqrt{x^{yz_{10}}}$ if $16$ across is $xyz_{10}+2^x,x\in\mathbb{N}$
15. The start of $20$a across${} - 10555_7$
13. $1$ across${} + 101042$ years ago
16. $3$ down reversed${} + 13010+{}$AP joined
5a. Undecimal $88840+3$ down
5b. Double $97-120$b down reversed${} + 61 $
20b. $4$ odd digits, or $5$ with flipped $19$b across
5c. A sum with the two summands${} - 4+6! $
17. AP joined, or a sum of the first $k$ primes

AP means Arithmetic Progression (of at least 3 terms)
Joined means concatenated

Text transcription of crossword here or see post source code

Hints for extraction:

I wonder why the clue numbers are labelled so weirdly... and what could fill in the blanks on the black spaces?

Perhaps the clues give a message? If only there were a convenient way to turn numbers into letters...

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  • $\begingroup$ Sorry, can't see the grid. Read my profile. Use Cubeupload. $\endgroup$
    – Scratch---Cat
    Apr 23, 2020 at 3:05
  • $\begingroup$ LaTeX missing space bar perhaps, use backslash+spacebar for displaying LaTeX spacebar $\endgroup$
    – Scratch---Cat
    Apr 23, 2020 at 3:07
  • $\begingroup$ @Scratch---Cat Sorry, I don't see where the missing spacebar is... $\endgroup$
    – boboquack
    Apr 23, 2020 at 7:57
  • $\begingroup$ @Scratch---Cat There is a text transcription of the crossword attached via pastebin in the post or you can click edit/improve this post and see it in the post's source code. $\endgroup$
    – boboquack
    Apr 23, 2020 at 8:23
  • $\begingroup$ this has been dormant for a while, any hint? :0 $\endgroup$
    – oAlt
    Jun 18, 2020 at 1:33

3 Answers 3

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Crossword

enter image description here

Across Clues

3. 13281 = Mixed 8, 13 and 21 = 13 × 1024 - √961
16. 410 = Joined 22, 11, and 02 = Joined 4 and 10 (prefix half of 1024)
12. 961 = 312 = 967 - 6
20a. 29557 = 11 × 2687 = 1722 - 27
8. 3747 = 3737 + 10 = 3796 - 72
19a. 3796 = Lao Digit Four in Unicode = (√9)796
21. 11 = Twice 1 = smaller prime factor of 29557 (11 × 2687)
1. 967 = 963 + 4 = 163rd prime
18. 64 = 82 = prefix 2/3 of 646
5d. 646 = Palindrome = 2/3 × (967 + 2)
19b. 973 = 7 × 139 = 963 + 10

Down Clues

3. 1024 = √410 = 210
15. 285 = 295 - 10 = 5557
13. 1977 = 967 + 1010 = 43 years ago!
16. 4331 = 4201 + 130 = 10 + 4321
5a. 1064 = 88811 = 40 + 1024
5b. 9796 = 9797 - 1 = 9735 + 61
20b. 5379 = 4 odd digits = Joined 5 and 379
5c. 716 = -4 + 720 (I couldn't match the first clue)
17. 963 = Joined 9, 6 and 3 = Sum of first 24 primes

However, I don't have any thought about the digits outside the grid. :(

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  • 1
    $\begingroup$ Nice job! (Sorry I didn't see this, my stackexchange inbox was playing up.) I'm going to post a hint for what the digits outside the grid are. $\endgroup$
    – boboquack
    Apr 28, 2020 at 0:05
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user65284 has already completed the crossword.

To obtain the final answer first do the following

Take each of the clue numbers and replace with a letter using the standard alphanumeric cipher (1=A, 2=B, ..., 26=Z). Reading left-to-right top the bottom we get the message COMPLETETHESQUARES or, adding spaces, COMPLETE THE SQUARES.

This suggests that the entries in the black squares should be filled in such a way that reading the numbers which intersect the black squares either down or across should yield a perfect square. To make this work, we need to also incorporate the digits outside the grid in the row or column in which they appear, so for example, the number in the 3rd row is 3XX29557XX0126096 which must be a perfect square when the Xs are filled in.

Under these conditions there is a unique way to complete the grid

enter image description here

And finally

Taking the numbers in the black squares in order and using the alphanumeric cipher once more we find that they spell out RESPOND, so that must be the message.

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  • $\begingroup$ Nice job - that's correct! Given that user65284 filled in the crossword, I'm going to award them the checkmark: hope you don't mind. :) $\endgroup$
    – boboquack
    Jun 18, 2020 at 23:02
  • $\begingroup$ @boboquack Yeah, no problem. Nice puzzle! $\endgroup$
    – hexomino
    Jun 19, 2020 at 0:38
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Partial answer:
5c. A sum with the two summands

A sum (7) with the two summands (1 and 6) $= 716 =$ −4+6 !

(The previous answer didn't say this, so l put it here.)

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  • $\begingroup$ CAPTCHA popped up again, sorry. $\endgroup$
    – Scratch---Cat
    Apr 29, 2020 at 1:21

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