WATERSPHERES

Below you will find 2 hints (to be used in order) for breaking into the encoded numbers in the Waterspheres puzzle, followed (after additional spoiler space) by a reveal of which letters stand for which numbers.

HINT 1:

Since there are 58 segments of water, the numbers across the top and the numbers down the side must each sum to 58.

Note that the 8 letters WATERSPH appear (in some order) both across the top and down the side -- they must stand for the digits 23456789 in some order. Since 2+3+4+5+6+7+8+9 = 44, the 3 "extra" letters across the top (E+R+E), and the 3 "extra letters down the side (R+S+T) must each sum to 58-44 = 14.

Since E+R+E = 14, then 14-R must be even, since 14-R = 2E, and therefore R must be even. So, one of these is true:

(R, E) = (2, 6) or (4, 5) or (6, 4) or (8, 3)

And since E+R+E = R+S+T, then S+T = 2E, which further narrows things down significantly.

And since A<W<P and none of those are consecutive, there are relatively few possibilities for what the remaining digits can be in each case. Examining these possibilities, and using some logic as to how the grid works, will help eliminate the wrong possibilities, leaving only the correct one.

Hint 2 appears below, after spoiler space:

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HINT 2:

From Hint 1, we know that (R, E) = (2, 6) or (4, 5) or (6, 4) or (8, 3).

Based on the given information, one of P and H must always be 9. Look at the P and H columns, and notice that 5 segments in each of the P and H columns require the corresponding Column 10 (the R near the end) segment to be filled (that is -- if that R is not filled, the water cannot reach the H or P columns). So, that immediately rules out R = 2, because if R = 2, then 3 out of those 5 sideways towers must have their PHER segments completely empty. That means that the P and H columns could have at most 8 segments filled, but since one of those letters must be the 9, this results in a contradiction.

Thus, (R, E) = (4, 5) or (6, 4) or (8, 3).

A similar, but slightly deeper, analysis eliminates the (8, 3) option, and another similar analysis (going a bit deeper, but still focusing mostly on the 5 sideways towers that span PHER), will knock out one of the remaining pairs.

A full reveal of the number that each letter stands for appears below after spoiler space:

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LETTERS DECODED:

The digits 2-9 are represented, in order, by: A, T, R, E, W, S, P, H

Happy Solving!