The grid contains the alphametic
SPRING + SUMMER + AUTUMN + WINTER = PRIEST in base 12,
using GSWETAUMIRNP for the digits 0 to 11 in order.
*The Four Seasons*, consisting of 12 movements in all,
are part of a set of concerti by Antonio Vivaldi, born on 4 March 1678.
He was nicknamed *il Prete Rosso* (the Red Priest) because of his red hair.

Conventions: * = anagram, < = reversal

No | Moved letter | Answer | Explanation |
---|---|---|---|

Across | |||

1 | Ro(P)e → P-eel | PEA | P(erfect) E(n) A(spic) |

8 | learner(S) → S-loth | AIL | AI + L |

12 | WILTON | (WILT + N) around O | |

14 | STRUMAE | (MATURE + S) * | |

15 | OARED | O + A + RED | |

16 | CRISP | CR + PSI < | |

17 | RING | NG after RI | |

18 | HANDCUFF | HAND + CU + FF | |

21 | HAN | H + N/A < | |

22 | ALAS | ALASKA−KA | |

24 | MERINO | MINER * + O | |

25 | GOSSAMER | GOER around MASS < | |

29 | ABLAUT | TUBA * around LA | |

32 | NEAR | EAR after N | |

33 | SUI | SUI(d) | |

34 | MISERERE | MISER + ERE | |

36 | WIN’T | W + IN + T | |

38 | ERATO | ERA + OT < | |

39 | OPTIC | OPT + I + C | |

40 | REVISAL | VIS in REAL | |

41 | NEARER | (A + R) in NE’ER | |

42 | (G)ratin → ratin-G | PER | P(i)E + R(ating) |

43 | (M)art → a-M-id | EAR | (insid)E AR(t) |

Down | |||

1 | PASCHAL | (A + L) after (C in PASH) | |

2 | ENTRALL | (c)ENTRALL(y) | |

3 | AGRIN | (AG(e) + RN) around I | |

4 | page(R) → Sp-R-ite | IMP | PIMP−P |

5 | fo(U)r → Po-U-nd | LARUMS | L + ARUMS |

6 | (A)new → w-A-ives | TE | TEN−N |

7 | we(T) → bas-T-e | OON | OO + N |

9 | AYRSHIRE | ((d)AIRY’S HER(d)) * | |

10 | ICE MAN | I + C + NAME < | |

11 | LADINO | LAID * + NO | |

13 | NAGOR | ORANG(e) * | |

19 | CUSUM | US in CUM | |

20 | FEMME | FÉE around MM | |

23 | AGLITTER | AG + LITTER | |

26 | unt(I)ed → un-I-ted | AUSTER | TEAR * around (U + S) |

27 | TARTANA | TARTAN + A | |

28 | AREOLAR | ARE + ORAL * | |

29 | ASTOOP | AS TO + OP | |

30 | BURPEE | RUB < + PEE | |

31 | ALWIN | A + L + WIN | |

35 | EASLE | (m)EASLE(d) | |

37 | Fro(W)stiness → W-rap | ICE | 2 meanings |

38 | see(N) → Easter-N | EVE | E(ggs) + V + E |

40 | (E)radiate → h-E-ad | RA | R(adiate) + (be)A(ms) |

This is one logical path for solving the alphametic. There are 12 clues with distinct moved letters (in 26dn either I or T could be moved, but T is the letter for 7dn), ie: 1ac P, 8ac S, 42ac G, 43ac M, 4dn R, 5dn U, 6dn A, 7dn T, 26dn I, 37dn W, 38dn N, 40dn E. Thus, the thematic number d is 12, the addition is in base 12, and the sum of moved letter + inserted letter for each entry is 11 (the letter for 0 being paired with the one for 11, 1 with 10, etc).

For the gap in 5dn, _LARUMS, the only letter that makes a new word is A, so the clue’s moved U gives us U + A = 11. Then for 6dn with moved A the inserted letter must be U, making UTE. Similarly, 26dn’s AUSTER_ can only become AUSTERE, so I + E = 11 and 40dn has I inserted to make RAI.

For the 6-letter word in the bottom row we have __IE__. Adding the letters that would make new words in the affected entries, the options are [MNPT][RS]IE[NRST][GNPRSTW], for which the only word with distinct letters is PRIEST (making PERP, ICER, EVES, TEAR). This establishes the remaining letter pairs as P+G, R+W, S+N and T+M and the top-row gaps can be completed as GWAUMN (making PEAG, WIMP, MOON, NAIL), so we know G < W < A < U < M < N. From pairings, we also know S < T < A < U < R < P.

A has at least four values below it and five above it, so it’s in the range [4, 6]. Similarly, U is in the range [5, 7]. Since A + U = 11 and A < U, A is one of {4, 5} and U is the corresponding element in {7, 6}. Since U < R < P, P must be at least 8. In the second column of digits, P + U + U + I must be at least 8 + 6 + 6 + I = 20 + I and there must be a carry of at least 1 to the leftmost column.

In the leftmost column we have S + S + A + W + (carry) = P = 11 − G,
so S + S + G + W + A < 11 (where S, W and A are all non-zero, being initial digits).
If G is 1 then the minimum sum is 2 + 2 + 1 + 3 + 4 = 12, too big;
if G is 2 or more, the minimum sum is 1 + 1 + 2 + 3 + 4 = 11, also too big;
so **G = 0** and **P = 11**.
As S < T < A, S is in the range [1, 3].
But if S is 3, the minimum sum is 3 + 3 + 0 + 1 + 5 = 12, too big; so S is one of {1, 2}.
As W < A, W is in the range [1, 4].
But if W is 4, the minimum sum is 1 + 1 + 0 + 4 + 5 = 11, too big; so W is one of {1, 2, 3}.
If W is 1, S must be 2; if W is 2, S must be 1;
if W is 3, S must be 1 (because 2 + 2 + 0 + 3 + 4 = 11, too big).

If (S, W) = (2, 1) then the pairings give (N, R) = (9, 10). In the rightmost column we have G + R + N + R = 0 + 10 + 9 + 10 = 29 = 2×12 + 5, giving T = 5 (so M = 6) and carrying (2). The sum in the fifth column is then 9 + E + 6 + E + (2), which is odd, so it can’t give S = 2 (modulo 12).

If (S, W) = (1, 3) then the pairings give (N, R) = (10, 8). In the rightmost column we have 0 + 8 + 10 + 8 = 26 = 2×12 + 2, giving T = 2 (so M = 9) and carrying (2). The sum in the fifth column is then 10 + E + 9 + E + (2) = 2E + 21; that needs to equal 12n + 1 (for some integer n ≥ 0), so 2E + 20 = 12n, or E = 6n − 10. If n = 2 then E = 2, but that value is already assigned to T; if n = 3 then E = 8, but that value is already assigned to R; other values of n give E < 0 or E > 11.

The only remaining possibility is **S = 1**, **W = 2**, and their pairs **N = 10**, **R = 9**.
The rightmost column gives 0 + 9 + 10 + 9 = 28 = 2×12 + 4,
so **T = 4** (and **M = 7**) and we carry (2).
That forces **A = 5** and **U = 6**, leaving only 3 and 8 for E and I, in some order.
In the fifth column, 10 + E + 7 + E + (2) = 12n + 1,
so E = 6n − 9, which is clearly a multiple of 3,
so **E = 3**, **I = 8** and the alphametic is complete.