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From D2, we see that the numerical values of the letters in D1, C2, E2 and D3 total 72, but from E1, D1 + E2 = 31. Therefore, C2 + D3 = 41. As C2's value is 24, D3's value must be 17, i.e Q.
The numerical value of A5 is 49. The only two possibilities for this total are 26 + 23 and 25 + 24. It cannot be the latter, because X and Y are consecutive letters.(In any case, X has already been used.) Therefore, the letters in A4 and B5 must be W/Z.
Now, look at C3, whose value is 87. X and Q (value 41) have already been placed, so B3 and C4 must equal 46. The only possibilities here are 26/20, 25/21 and 24/22. As Z and X are already in place, the values must be 25/21 (Y/U). Y cannot be adjacent to X, so Y must be in C4 and U in B3. From this, Z must be in A4 and W in B5.
From C5, D5 must be R. This places V in E4; K in E2; T in D1; C in B1 and S in A2.
Now look at D1 and E2. The difference in their values is 7. The letters in D2 and E1 are common to both of these cells. Therefore, the value of the letter in E3 must be seven greater than the one in C1.
Similarly, the difference between C1 and A3 is 2; the difference between A3 and C5 is 3.
So, let C1 be y; A3 be y + 2; C5 be y + 5 and E3 be y + 7.
The remaining, unused, letters are A, B, D, E, F, G, H, I, J, M, N, O and P.
C1 cannot be B or D, so the five possibilities are E, F, G, H and I.
+-------+---+---+---+---+---+
| y | E | F | G | H | I |
+-------+---+---+---+---+---+
| y+2 | G | H | I | J | K |
+-------+---+---+---+---+---+
| y+5 | J | K | L | M | N |
+-------+---+---+---+---+---+
| y+7 | L | M | N | O | P |
+-------+---+---+---+---+---+
As K has already been used, and L does not feature in the grid, the only valid option is H/J/M/O.
The letter values in A1 and B2 must now total 8. The only possibilities are A/G and B/F, but B cannot appear in A1 or B2, so it has to be A/G. G cannot appear in B2, so it must be A, with G in A1.
The letter values in D4 and E5 must now total 16. The only possibility here is B/N. N cannot appear in D4, so it must be B, with N in E5.
The letter values in A5 and B4 must now total 21. The only possibility here is P/E. The letter values in D2 and E1 must now total 10. The only possibility here is D/F. This places the only remaining letter (the I) in the central cell.
The "25" in B3 places E in B4 and P in A5 ( 25 - (10 + 1 + 9)). The "30" in D3 places D in D2 and F in E1 ( 30 - (9 + 2 + 15)).
Hence:
+---+---+---+---+---+
| G | C | H | T | F |
+---+---+---+---+---+
| S | A | X | D | K |
+---+---+---+---+---+
| J | U | I | Q | O |
+---+---+---+---+---+
| Z | E | Y | B | V |
+---+---+---+---+---+
| P | W | M | R | N |
+---+---+---+---+---+
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