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 ==Table==   ==Table== 
−  # D = 1,2
 
−  # A = 3,4
 
−  # N = 4
 
−  # I = 1
 
−  # E = 1, (9 or 10)
 
−  # S = 1
 
−  # P = 2
 
−  # R = 3
 
−  # Y = 1
 
− 
 
−  ==Reasoning (NEW)==
 
−  # 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.
 
− 
 
 { border=1   { border=1 
  1  D  S  I  E    1  D  S  I  E 
Line 31: 
Line 9: 
  2  P  Y  E  D    2  P  Y  E  D 
     
−   3  R   +   3  R  A  E  S 
 +   
 +   4  A  N  B  T 
 +   
 +   5  D  I  R  
 +   
 +   6  R  O  W  N 
 +   
 +   7  A  M  E  G 
 +   
 +   8  A 
 +   
 +   9  S 
 +   
 +   10  P  R  T  E 
 +   
 +   J  I  R  L  E 
 +   
 +   Q  N  O  T  V 
     
−   4  A   +   K  C  U  R  Y 
 }   } 
   
−  ==Reasoning==
 
   
−  # 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.
 +  Another 70 pointer is 10JQK101010 = '''trouper''' 
−  # 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 4sequence, 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 4sequence.
 
−  ### 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.
 
−  # 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.
 