Alice

Puzzle_Boat

[1]

Table

1. D1 = 1
2. A1 = 2
3. N1 = 3
4. D2 = 4
5. I1 = 1
6. E1 = 1
7. S1 = 1

Reasoning

1. 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.
2. DANDIES=13. Speculate:
   1. Assume that we have a 4-sequence.
      1. Assume that the sequence is first. Then D1=1,A1=2,N1=3,D2=4,I1=1,E1=1,S1=1.
      2. Apply this information to PRAISED=13.
         1. 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.