## Letter Lines

The letters A-I represent the digits 1-9 in some order. The letters are placed in a line from left to right. Determine which digit is represented by each letter and the order of the letters from left to right, given the following clues. ("A _ 6" means "A is two positions to the left of 6." "_ _ 8" means "There are at least two positions to the left of 8.")
 Puzzle 1 A _ _ _ 5 4 _ _ B C D _ _ _ 2 6 _ _ _ E 7 _ _ _ _ F 5 _ G _ _ _ 8 _ H 1 _ I _ 9 Puzzle 2 A _ 6 B _ _ _ 4 C _ _ _ _ 7 1 _ D _ 7 1 _ _ _ _ E 9 _ F G _ _ 8 2 _ _ H 5 _ _ _ I

Source: Original. [Has anyone seen this sort of puzzle before? It seems new to me.]

Solutions were received from Denis Borris, Philippe Fondanaiche. I didn't get a logical approach, but here are the correct answers:

```Puzzle 1:
C A D B H I F G E
4 7 8 1 6 5 2 9 3
or
4 6 8 7 1 5 2 3 9
C A D B H E I G F

Puzzle 2:
G C F B D I A E H
9 5 1 8 3 2 7 4 6
```

Update 4/30/01: Hareendra Yalamanchili sent logical solutions:
1. ```Logic for puzzle 1

From A---5 and 5-G- we have A---5-G-
G is at spot 7 or 8.
The first 2 spots must be A,C,or D and the last 2 are 2,3,or 9

1-I-9 has 5 possibilities:
1.  |  1 | A  | I  |    |   9 |  5 |    |  G |    |
F has 2 possibilities:
1a.|  1 | A7 | I  |    |   9 |  5 | F  |  G |    |
leaves no spot for E
1b.|  1 | A  | I  |  7 |   9 |  5 |    |  G |  F |
forces H
|  1 | A  | I8 |  7 |  H9 |  5 |    |  G |  F |
forces E
|  1 | A6 | I8 |  7 |  H9 | E5 |    |  G |  F |
leaves no spot for B.

2.  | A  |  1 |    | I  |   5 |  9 |    | G  |    |
2 and 3 must be at the last 2 spots, forcing
| A  |  1 |    | I  |  D5 |  9 |    | G3 |  2 |
forces E
| A  |  1 |  6 | I  |  D5 |  9 | E  | G3 |  2 |
leaves no space for B

Since I is not at spots 3 or 4 this means that the last 5 spots are
E,F,G,H, and I
This forces B to spot 4.
|  4 |    |    | B  |    |    |    |    |    |
D cannot be at spots 4 or 5, so 2 is not at spot 8 or 9, making the last
2 spots 3 and 9, thus eliminating case 3.

4.  |  4 |    |    | B1 |    | I  |    |  9 |  3 |
F&G can only be in the 7th and 8 spots, forcing E and then H
|  4 |    |  8 | B1 | H6 | I  |    |  9 | E3 |
forces A---5-G-
|  4 | A  |  8 | B1 | H6 | I5 |    | G9 | E3 |
giving a solution
| C4 | A7 | D8 | B1 | H6 | I5 | F2 | G9 | E3 |

5.  |  4 |    |    | B  |  1 |    | I  |  3 |  9 |
forces A---5--G
|  4 | A  |    | B  |  1 |  5 | I  | G3 |  9 |
forces D,E,F and gives the solution
| C4 | A6 | D8 | B7 | H1 | E5 | I2 | G3 | F9 |
```
2. ```Puzzle 2
A _ 6
B _ _ _ 4
C _ _ _ _ 7
1 _ D _ 7
1 _ _ _ _ E
9 _ F
G _ _ 8
2 _ _ H
5 _ _ _ I

Note the the first two must be A,B,C, or G
The C-E hints give us

|  C | 1  |   | D  |   | 7  |  E   |
Which can fit in 3 ways
1.|    |    | C  | 1  |    | D  |    | 7  | E  |
Then G| | |8 can fit in 2 places.
1a. |    | G  | C  | 1  |  8 | D  |    | 7  | E  |
This forces B into
|    | G  | C  | 1  | B8 | D  |    | 7  | E4 |
and A must be on the left
| A  | G  | C6 | 1  | B8 | D  |    | 7  | E4 |
leaving no place for I.

1b. |    |    | C  | 1G |    | D  |  8 |  7 | E  |
Then A&B must be on the left,
| A  | B  | C6 | 1G |    | D4 |  8 |  7 | E  |
forces F
| A  | B  | C6 | 1G |  9 | D4 | F8 |  7 | E  |
leaves no space for H&I

2.   | C  | 1  |    | D  |    | 7  | E  |    |    |
b---4 fits in 2 ways, and the 2nd spot is either A or G
2a. | C  | G1 | B  | D  |  8 | 7  | E4 |    |    |
forces A&H into the same square
2b. | C  | A1 | B  | D6 |    |  7 | E4 |    |    |
forces F and forces the last square to be I
| C  | A1 | B9 | D6 | F5 |  7 | E4 |    | I  |
forces G&H into the same square
2c. | C  | A1 |    | D6 | B  |  7 | E  |    |  4 |
leaves no square for G
2d. | C  | G1 |    | D  | B8 |  7 | E  |    |  4 |
The 3rd and last squares cannot be A, H, or I forcing
F into both squares
3.   |    | C  | 1  |    | D  |    | 7  | E  |    |
b---4 fits in 2 ways
3a. | B  | C  | 1  |    | D4 |    | 7  | E  |    |
forces the last square to be H
| B  | C  | 1  |    | D4 |  2 | 7  | E  | H  |
forces G
| B  | C  | 1  |    | D4 | G2 | 7  | E  | H8 |
leaving no space for A
3b. |    | C  | 1  | B  | D  |    | 7  | E4 |    |
forces the first square to be G
| G  | C  | 1  | B8 | D  |    | 7  | E4 |    |
forces H
| G  | C  | 1  | B8 | D  |  2 | 7  | E4 | H  |
forces I
| G  | C5 | 1  | B8 | D  | I2 | 7  | E4 | H  |
yielding the solution
| G9 | C5 | 1F | B8 | D3 | I2 | A7 | E4 | H6 |
```

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