- D = 1,2
- A = 3,4
- N = 4
- I = 1
- E = 1, (9 or 10)
- S = 1
- P = 2
- R = 3
- Y = 1
- TROUBLE=67. Clearly, E>=7. Thus E in DANDIES is different from the E in TROUBLE. In fact, E=1,2,3. E cannot be 4 since then I+E+S = 12. Assume E=2. Then I+E+S=6=>D+A+N+D=7. If the second D=2, then D+A+N=5, which is impossible (since its a sequence). Likewise, no sequence makes D+A+N+D=7.
- Assume E=3. Then I+E+S=9 => D+A+N+D= 4. Then D,A,N,D=1. Applied to PRAISED, S=E=D=1, contradicting E=3.
- Thus E=1. And S,I = 1 as well. D+A+N+D=10.
- Apply to SPRAYED = 16. Again, the E cannot be TROUBLE's E. So here, E=1 => Y=1, D=1. Contradiction!. Clearly each letter can appear more than twice. Doh!
- First point still holds. E=1 or 3. If E=3 then I+S+E=9, and D+A+N+D=4. => D,A,N,D=1. There should be NO MORE 1s. Now to PRAISED. Again, by the same reasoning, the E in PRAISED must be 1 or 3. Since we've run out of 1s, E there must be 3, implying that P+R+A+I=4. But this means that P,R,A=1, a contradiction!
- E=1 and S,I=1. D+A+N+D=10. One of the Ds must be equal to 1. D,S,I,E =1. There should be NO MORE 1s on the board. Move to PRAISED. By the same logic, E=1 or 3. Since there can be no more 1s, we must have that S=E=D=1. P+R+A+I=10 which is fine since I=1 and P,R,A=2,3,4. By before, D,A,N=2,3,4 or A,N,D=2,3,4. Either way, we now have all 1s accounted for and 2 of 2,3,4 each accounted for.
- For SPRAYED=16, E<=4. E cannot be 5 since Y+E+D=15. If E=4 then Y+E+D=12 and S+P+R+A = 4. Since they would all have to be 1s, we would have a contradiction. If E=1 then Y=1, a contradiction.
- If E=3 then Y+E+D=9. Then S+P+R+A=7. They cannot form a sequence, thus A must go with Y,E,D. If Y,E,D is a triplet, then S+P+R=4, which is impossible. If Y,E,D is a sequence, then A=1 which is also impoosible.
- So E=2. Then Y+E+D=6 and S+P+R+A = 10. Y,E,D must form a triplet lest Y=1. If we let A=2, then S+P+R=8. But 1+2+3 = 6 and 2+3+4=9. So we must have a 4 sequence. S,P,R,A=1,2,3,4. So we'd have Y,E,D=2. From before, P=2. All 2s are accounted for.
- Back to DANDIES. D,A,N=2,3,4 or A,N,D=2,3,4. But A cannot be 2, so D=2, A=3, N=4.
- ANDRIES. E=1,2,3,4,5. E=5 => A+N+D+R=6 => A+N+R >= 5. But since A,N,R>=3, A+N+R>=9. If E=4, then A+N+D+R=9. A+N+D+R cannot be a sequence since it's >= 10. A+N+D+R cannot be 4 of a kind. Thus, R must go with the second half. R!=2. So then R=4. Then A+N+D=5 which is impossible.
- If E=3... Since I!=2 we must have I=3 and S=3. But that is a contradiction.
- If E=2 and since I!=2 we must have I=1,E=2,S=3. A+N+D+R=15. 4*b+6=15 => 4*b=9. Impossible.
- E=1. A+N+D+R=18. Since R!=1, this must be a sequence. Only option is A=3,N=4,D=5,R=6.
- TROUBLE=67. For a 4 sequence, b,b+1,b+2,b+3,x,x,x; b=10 => x = 9; b=9 impossible; b=8 => x = 10. For a 3 sequence, b,b+1,b+2,x,x,x,x; b=10 impossible; b = 9 impossible; b = 8 => x=10. E1 = 9 or 10.
- If a sequence of 3, then the smallest the base of the sequence, b, must be b+(b+1)+(b+2) +1+1+1+1 <= 13. Thus, 3*b+7 <= 13. b <= 2. If it's a 4-sequence, then we have b+(b+1)+(b+2)+(b+3)+1+1+1 <= 13. 4*b+9 <= 13. b <= 1.
- DANDIES=13. Speculate:
- Assume that we have a 4-sequence.
- Assume that the sequence is first. Then D1=1,A1=2,N1=3,D2=4,I1=1,E2=2,S1=1.
- Apply this information to PRAISED=13.
- If D=D2, then the sequence must end there, and we must have 1+1+1+1+2+3+4. Giving us P1=1,R1=1,A2=1,S2=2,E2=3. A contradiction (because of E).
- Then D=D1. Then the last three must be 1 => E2=1. A contradiction.
- Pop. Assume that the sequence is second. Then D1=1,A1=1,N1=1,D2=1,I1=2,E2=3,S1=4. Again, applying to praised yields the same contradiction that E2=1.
- Pop. We must have a 3 sequence.
- Assume that b=1. Then 4*x+3*b+3 = 13. 4*x=7. Impossible.
- Assume that b=2.
- Assume that the sequence is second. Then I1=2,E2=3,S1=4,D1=1,A1=1,N1=1,D2=1. But when applied to PRAISED, E2=1 a contradiction.
- Assume that the sequence is first. Then D1=2,A1=3,N1=4, D2=1, I1=1, E2=1,S1=1.
- Apply to PRAISED. No matter what, the triplet must be last then. If D=D1, then E2=2 and we have a contradiction. Thus, D=D2, E2=1.
- Assume that we have a 4-sequence.
- Popping since the above is correct. PRAI = 10. If this is a sequence, then P=1, R=2,A=3, I=4. But we already have 4 1's, so this is not the case. Therefore, the sequence is PRA and I is part of a quartet. P+P+1+P+2 = 9 => 3*P = 6 => P=2, R=3,A=4.