7
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Group words in pairs and find connection with common answer.

third heart
chapters final
squares length
actinium immemorial
A0 bible
sodium sum
time lion
minor revelation
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  • $\begingroup$ Should the first word in the fourth row, which is currently actinum, be actinium? $\endgroup$
    – Gareth McCaughan
    Jan 1 at 2:22
  • 1
    $\begingroup$ Also, are we supposed to understand that each pair is one word from the left column and one from the right column? (Of course you might well prefer not to answer that question.) $\endgroup$
    – Gareth McCaughan
    Jan 1 at 2:23
  • $\begingroup$ Thanks for the correction. And, no, it's just sixteen words happened to be arranged in a grid. $\endgroup$
    – David
    Jan 1 at 2:43

1 Answer 1

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The pairs are:

TIME / IMMEMORIAL. In English law, "time immemorial" (as distinguished from historical time) actually has a formal ending date, in the year 1189.
LION / HEART. The year 1189 is when Henry II was replaced as king of England by Richard I, nicknamed "the Lionheart".
BIBLE / CHAPTERS. There are 1189 chapters in the (Protestant) Bible.
A0 / LENGTH. A sheet of A0 paper is 1189mm long.
SODIUM / ACTINIUM. Both elements, with atomic numbers 11 and 89 respectively.
MINOR / THIRD. The name of a musical interval, which in equal temperament corresponds to a frequency ratio of about 1.189.
SUM / SQUARES. The number 1189, being the product of two prime numbers that are 1 mod 4, is a sum of two squares (in two different ways). (For those who prefer more constructive proofs: it's 10^2+33^2 and 17^2+30^2.)
FINAL / REVELATION. The final chapter of the biblical book of Revelation is the 1189th chapter in the Bible. (This is really the same fact as for BIBLE CHAPTERS above, since Revelation is the last book of the Bible.)

So the common answer is clearly

1189.

It seems likely that

there's some particular reason why this is appropriate for New Year's Day of 2023 (given the title of the puzzle), and indeed David confirms in comments that there is, but I haven't yet figured out what it is.

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  • $\begingroup$ The element you found confusing is perhaps a flaw in the puzzle as you had the right idea. There is a connection with the day but isn't part of the puzzle as stated. Contemplating whether proof of existence of a solution is a solution for one of the elements. I think technically it is. $\endgroup$
    – David
    Jan 1 at 3:01
  • $\begingroup$ OK, adjusted answer appropriately. Still thinking about connections with the day. $\endgroup$
    – Gareth McCaughan
    Jan 1 at 3:07

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