Solution to Tryst of Fate

As hinted by the italicized words slant puzzle in the flavor text, this is a Slant puzzle, the solution of which will always form paths that connect points/lovers on the outer edge pairwise. The given numbers alone are not sufficient to solve the puzzle, however, and one must use the additional information that for the two people to meet at the middle of the path, the number of steps they walk differ by at most 1 (depends on the parity of the path length), and that these meeting points are already given. The completed logic puzzle looks like:

	0		1		2		1		1		1		1		1		0		1
						╲		╲		╲		╲		╲				╱
	2		1		2		1		1		2		3		1		2		3	1
		╲		╱		╲		╱		╱				╱		╱		╱
	0		4		1		2		3		1		3		1		2		1
				╲				╱		╲		╲				╱		╱
	2		1		1		3		1		2		2		3		2		1
				╲		╱		╱		╱		╱		╱		╱
	0		2		4		2		1		3		1		3		2		0
		╱		╱		╲		╲						╱		╲		╲
	1		2		2		1		3		1		3		2		1		2
		╱		╱		╲		╱		╱				╱		╲		╱
	1		3		1		2		3		2		2		1		2		1
		╱				╲		╱		╲		╲				╲		╱
	1		1		3		3		1		1		3		2		2		1
		╱		╱				╲		╲		╱		╲		╲
	1		2		2		2		2		3		1		1		4		0
		╱				╲		╲		╲		╲		╲				╲
	1		1		0		1		1		1		1		2		0		1

The next step is to index into each person’s name by the number of steps they took to reach the meeting point, and read the extracted letters clockwise around the garden, separated by pronouns. This is hinted by “The women then named…” and “The men named…”. We would then get

This reveals that the women’s “bets matrimonial” refers to Katie Baker’s Matrimonial Moneyball, a scoring system for NYT wedding announcements. The checksums given for “the eight letters” refers to the points given to each of NUPTIALS categories (Names, Universities, Parents, Tropes, Identifiers, Avocations, Locales, and Special Situations) described by Katie Baker if we summed up the scores across all couples. Pairing up the bios for each couple, we can score their profile with this system and get a number for each couple.

For the men’s part, the “degree-ten polynomial” turns out to be the characteristic polynomial. For an n×n square matrix, its characteristic polynomial will be degree n, so we need to find a 10×10 matrix. While the hedge garden is 9×9, it has 10×10 vertices (the small square hedges, where some of the Slant clues were provided). Using the Slant solution from earlier, we can backfill a number for each of the vertices (by counting how many diagonal hedges end at each node) to get a 10×10 matrix, shown in the yellow numbers above. We can then find the characteristic polynomial of this matrix, which turns out to be:

P(x) = x^10 − 16x^9 − 20x^8 + 18x^7 + 365x^6 + 249x^5 − 223x^4 + 35x^3 − 359x^2 − 50x

Finally, the flavor text hints that the polynomial “argued,” i.e. took as argument, the values of each pair (the scores from Matrimonial Moneyball), and that we need to mod the final result by 31. Using 1=A, 2=B, etc., this gives a letter for each couple. Placing each couple’s letter on their corresponding meeting point and read from top to bottom yields Cheers's Hester Crane, which is portrayed by NANCY MARCHAND, the final answer. (Incidentally, Hester’s middle name is Rose!)

For reference, the tallied scores for each couple are below. Further breakdowns for each N/U/P/T/I/A/L/S scores are in in each cell's title attribute (mouse users: hover on a NUPITALS score to view its breakdown).

The couples’ letters on their meeting point:

	0		1		2		1		1		1		1		1		0		1
		C		H		╲		╲		╲		╲		╲		E		╱
	2		1		2		1		1		2		3		1		2		3	1
		╲		╱		╲		╱		╱		E		╱		╱		╱
	0		4		1		2		3		1		3		1		2		1
		R		╲		S		╱		╲		╲		S		╱		╱
	2		1		1		3		1		2		2		3		2		1
		H		╲		╱		╱		╱		╱		╱		╱		E
	0		2		4		2		1		3		1		3		2		0
		╱		╱		╲		╲		S		T		╱		╲		╲
	1		2		2		1		3		1		3		2		1		2
		╱		╱		╲		╱		╱		E		╱		╲		╱
	1		3		1		2		3		2		2		1		2		1
		╱		R		╲		╱		╲		╲		C		╲		╱
	1		1		3		3		1		1		3		2		2		1
		╱		╱		R		╲		╲		╱		╲		╲		A
	1		2		2		2		2		3		1		1		4		0
		╱		N		╲		╲		╲		╲		╲		E		╲
	1		1		0		1		1		1		1		2		0		1