1.) A1 tells you that Aries and Aquarius are in D1 and A4 or vice versa. It also tells you that one of them is directly above an X. So D2 must be X.
2.) The shortest sign is Leo. The signs containing an S are Aries, Taurus, Scorpio, Sagittarius, Aquarius, Pisces. You know from step 1 that the S sign not adjacent to Leo must be Aries or Aquarius. Therefore Leo must be adjacent to either D1 or A4, as well as adjacent to 4 other S signs. But we also know that Leo must be adjacent to Libra. Being adjacent to 6 signs rules out all cells except B2, B3, C2, and C3. B2 and C3 are not adjacent to either corner, so they cannot be adjacent to all but one sign containing an S. C2 cannot be just left of Libra. Therefore B3 is Leo.
3.) According to D2, Libra must be in C3.
4.) The half vowel signs are Gemini and Taurus. D2 tells you that Taurus is adjacent to Leo. Therefore, according to C3, C1 is Gemini.
5.) According to C1, B4 and C4 are in alphabetical order. If D4 were to be Virgo, then Row 4 would definitely be in alphabetical order, since A4 starts with A (step 1). B3 tells you that only one row or column can be in alphabetical order (not counting X’s). So if Row 4 is in alphabetical order, nothing else can be. But if Virgo is D4, then Column D would also be in alphabetical order. You know that D1 is the first or second sign alphabetically, with the other being in A4. Virgo is dead last, and D2 is X. So regardless of what D3 is, if D4 is Virgo, Column D would be in alphabetical order. B3 says only one row or column is in alphabetical order, and it ends in Virgo. Therefore Virgo cannot go in D4, or one row and one column would be in alphabetical order.
Column A ends in Aquarius or Aries; it can’t be in alphabetical order.
Column B has Pisces or Sagittarius at the end (C1); it can’t be the one in alphabetical order.
Column C has Scorpio or Taurus at the end (C1); it can’t be the one in alphabetical order.
Column D was just ruled out above because it would mean Row 4 is also alphabetical, ending in Virgo.
Row 1 ends in Aquarius or Aries; it can’t be in alphabetical order.
Row 2 does not end in Virgo; it can’t be the one in alphabetical order.
Row 4 is ruled out for the same reason as Column D.
Row 3 is the only option left, meaning Virgo is D3.
6.) If C2 is X, Column C would be in alphabetical order (C1), but the alphabetical row/column is Row 3. So C2 must contain a sign. B2 also must contain a sign because it ends in Pisces/Sagittarius (C1) and begins with either X or Capricorn or Cancer. You know that because all the signs containing an S must be adjacent to B3 and all the other non-S signs are used. So if neither B2 nor C2 can be X, according to D3, A2 is X.
6.) D4 cannot be an X or Row 4 will be in alphabetical order: Aquarius/Aries (A1), Pisces/Sagittarius (C1), Scorpio/Taurus (C1), X. We established why B2 and C2 cannot be X in the other step 6 above. Knowing that D1 is a sign, the remaining X must be A3 or B1, but A1 says no column can have a third X. Therefore B1 is X.
7.) Take a look at B1. The only way for all signs ending in -us to be in the same row is if A4 is Aquarius, B4 is Sagittarius, and C4 is Taurus. Nothing prevents that scenario. The only way for any sign to be adjacent to all the 5-letter signs is if that sign is C2 and Aries is D1. Nothing prevents that scenario either. However, in the first scenario, if Aquarius is A4, then Aries is D1. In the second scenario, Aries is D1. Either way, Aries is D1.
8.) According to A1, Aquarius is A4.
9.) You still didn’t know which of B1’s scenarios is true. Now A4 tells you that it’s the second one, meaning Henry’s favorite sign is C2. That doesn’t help at all, but it also means that the 3 -us signs are not in the same row. So if B4 is Sagittarius, then C4 cannot be Taurus. But if B4 is Sagittarius, C4 cannot be Scorpio either, according to A2. So B4 cannot be Sagittarius, since those are the only two options for C4, according to C1, which therefore tells you that B4 must be Pisces.
10.) Row 3 is the one in alphabetical order and all the X’s are placed. So A3 has to be Cancer or Capricorn. The other one must be in D4, because all signs ending in S except Aries are adjacent to Leo (D2). So B2 and C2 and C4 are some combination of Scorpio, Taurus, and Sagittarius, C4 being one of the first two (C1). If C4 is Taurus, then B2 and C2 would both begin with S, and that cannot be, according to A2. Therefore, according to C1, C4 is Scorpio.
11.) A2 and D2 are both X’s. This means B2 must be after C2 alphabetically, because there can only be one row or column in alphabetical order and it has to end with Virgo (B3). That row/column is Row 3. So since B2 and C2 are narrowed down to Sagittarius and Taurus, you know Taurus is B2 and Sagittarius is C2.
12.) D4 is adjacent to 3 signs. Half of 3 is 1.5. A3 is adjacent to 4 signs. Half of 4 is 2, a whole number. Therefore C2 tells you that A3 is Cancer.
13.) And Capricorn is left for D4.