Let’s continue the story from a previous post.

Since *ONE + ONE = TWO* we can hope to constraint the values of *N* to the value of *W* somehow. For starters we can say that because of the rightmost column when *E>4* then *W* is odd because of the carry, while *W* is even when *E<5*

```
1
ONE +
ONE = when E is greater than 4
-----
TWO
0
ONE +
ONE = when E is less than 5
-----
TWO
```

In a table this becomes:

E | W |
---|---|

1 | even (0,2,4,6,8) |

6 | odd (1,3,5,7,9) |

2 | even (0,2,4,6,8) |

7 | odd (1,3,5,7,9) |

Likewise, there is a similar relationship linking *T* and *N*: if *T* is odd there must be some carry from the nearby column, so that *N>4*, while *T* even implies *N<5*. In a table

T | N |
---|---|

4 | (0,1,2,3,4) |

5 | (5,6,7,8,9) |

8 | (0,1,2,3,4) |

9 | (5,6,7,8,9) |

Notice that there are conflicts: we will resolve them sticking these minitables into the master table we are maintaining, in a moment. When *W* is even (no carry from *E + E*) the rightmost digit of *N + N* is *W* (when *W* is odd we must decrease it by 1 of course). A similar argument can be made for the latter table. Since we know that *R = 0* we can drop the cases where either *W* or *N* are 0, or those where *W = N*. This leads to the table

W | N |
---|---|

2 | 1 |

2 | 6 |

4 | 2 |

4 | 7 |

6 | 3 |

6 | 8 |

8 | 4 |

8 | 9 |

1 | 5 |

3 | 1 |

3 | 6 |

5 | 2 |

5 | 7 |

7 | 3 |

7 | 8 |

9 | 4 |

Were we to inject this table – which is quite long – into the master table that would make it too long to be manageable by hand; or not, depending on your laziness: it saturates mine for sure, so we have to find some smarter trick. The following table could be a starting point, we add columns to list the *possible* values for *W*

R | O | T | I | E | W | spare (including W) |
---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | even (2,4,6,8) | 36789 |

0 | 2 | 4 | 5 | 6 | odd (1,3,5,7,9) | 13789 |

0 | 2 | 5 | 4 | 1 | even (2,4,6,8) | 36789 |

0 | 2 | 5 | 4 | 6 | odd (1,3,5,7,9) | 13789 |

0 | 4 | 8 | 9 | 2 | even (2,4,6,8) | 13567 |

0 | 4 | 8 | 9 | 7 | odd (1,3,5,7,9) | 12356 |

0 | 4 | 9 | 8 | 2 | even (2,4,6,8) | 13567 |

0 | 4 | 9 | 8 | 7 | odd (1,3,5,7,9) | 12356 |

We now exclude the values of *W* that conflict with the other letters, so in example in the first row

R | O | T | I | E | W | spare (including W) |
---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | even (2,4,6,8) | 36789 |

*W* cannot have the values 2 or 4, so we get

R | O | T | I | E | W | spare (including W) |
---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | even (6,8) | 36789 |

Now we add the column for *N* starting from the values for *W*

R | O | T | I | E | W | N | spare (including W and N) |
---|---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | even (6,8) | (3,8;4,9) | 36789 |

We can exclude the conflicts here too

R | O | T | I | E | W | N | spare (including W and N) |
---|---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | even (6,8) | (3,8;9) | 36789 |

Then we get the new table

R | O | T | I | E | W | N | spare (including W and N) |
---|---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | even (6,8) | (3,8;9) | 36789 |

0 | 2 | 4 | 5 | 6 | odd (3,7) | (1,6;3,8) | 13789 |

0 | 2 | 5 | 4 | 1 | even (6,8) | (3,8;9) | 36789 |

0 | 2 | 5 | 4 | 6 | odd (3,7) | (1,6;3,8) | 13789 |

0 | 4 | 8 | 9 | 2 | even (6) | (3) | 13567 |

0 | 4 | 8 | 9 | 7 | odd (1,3,5) | (5;1,6;2) | 12356 |

0 | 4 | 9 | 8 | 2 | even (6) | (3) | 13567 |

0 | 4 | 9 | 8 | 7 | odd (1,3,5) | (5;1,6;2) | 12356 |

Now since

T | N |
---|---|

even | (0,1,2,3,4) |

odd | (5,6,7,8,9) |

we obtain

R | O | T | I | E | W | N | spare (including W and N) |
---|---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | even (6) | (3) | 36789 |

0 | 2 | 4 | 5 | 6 | odd (3,7) | (1;3) | 13789 |

0 | 2 | 5 | 4 | 1 | even (6, 8) | (8;9) | 36789 |

0 | 2 | 5 | 4 | 6 | odd (7) | (8) | 13789 |

0 | 4 | 8 | 9 | 2 | even (6) | (3) | 13567 |

0 | 4 | 8 | 9 | 7 | odd (3, 5) | (1;2) | 12356 |

0 | 4 | 9 | 8 | 2 | even (6) | () | 13567 |

0 | 4 | 9 | 8 | 7 | odd (1, 3) | (5;6) | 12356 |

(Second to last row is empty, so we can drop it)

Now the master table becomes

R | O | T | I | E | W | N | spare |
---|---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | 6 | 3 | 789 |

0 | 2 | 4 | 5 | 6 | 3 | 1 | 789 |

0 | 2 | 4 | 5 | 6 | 7 | 3 | 189 |

0 | 2 | 5 | 4 | 1 | 6 | 8 | 379 |

0 | 2 | 5 | 4 | 1 | 8 | 9 | 367 |

0 | 2 | 5 | 4 | 6 | 7 | 8 | 139 |

0 | 4 | 8 | 9 | 2 | 6 | 3 | 157 |

0 | 4 | 8 | 9 | 7 | 3 | 1 | 256 |

0 | 4 | 8 | 9 | 7 | 5 | 2 | 136 |

0 | 4 | 9 | 8 | 7 | 1 | 5 | 236 |

0 | 4 | 9 | 8 | 7 | 3 | 6 | 125 |

Now We extract the last bit of information from the statement of the puzzle:

```
FOU +
ON =
-----
FIV
```

Looking the rightmost column we see that

a. the rightmost digit of *U+N* equals *V*

b. when *I* is even there is no carry, hence *U+N < 10*, on the other hand when *I* is odd *U+N > 9*

Let’s augment the master table with the possible values for the sum of *U* and *V*

R | O | T | I | E | W | N | spare | U+N | constraint |
---|---|---|---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | 6 | 3 | 789 | (10,11,12) | U+N>9 |

0 | 2 | 4 | 5 | 6 | 3 | 1 | 789 | (8,9,10) | U+N>9 |

0 | 2 | 4 | 5 | 6 | 7 | 3 | 189 | (4,11,12) | U+N>9 |

0 | 2 | 5 | 4 | 1 | 6 | 8 | 379 | (11,15,17) | U+N<10 |

0 | 2 | 5 | 4 | 1 | 8 | 9 | 367 | (12,15,16) | U+N<10 |

0 | 2 | 5 | 4 | 6 | 7 | 8 | 139 | (9,11,17) | U+N9 |

0 | 4 | 8 | 9 | 7 | 3 | 1 | 256 | (3,6,7) | U+N>9 |

0 | 4 | 8 | 9 | 7 | 5 | 2 | 136 | (3,5,8) | U+N>9 |

0 | 4 | 9 | 8 | 7 | 1 | 5 | 236 | (7,8,11) | U+N<10 |

0 | 4 | 9 | 8 | 7 | 3 | 6 | 125 | (7,8,11) | U+N<10 |

We must enforce the constraint, obtaining

R | O | T | I | E | W | N | U+N | U | V |
---|---|---|---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 1 | 6 | 3 | (10,11,12) | (7,8,9) | (0,1,2) |

0 | 2 | 4 | 5 | 6 | 3 | 1 | (10) | (9) | (0) |

0 | 2 | 4 | 5 | 6 | 7 | 3 | (11,12) | (8,9) | (1,2) |

0 | 2 | 5 | 4 | 6 | 7 | 8 | (9) | (1) | (9) |

0 | 4 | 8 | 9 | 2 | 6 | 3 | (10) | (7) | (0) |

0 | 4 | 9 | 8 | 7 | 1 | 5 | (7,8) | (2,3) | (7,8) |

0 | 4 | 9 | 8 | 7 | 3 | 6 | (7,8) | (1,2) | (7,8) |

If we eliminate the conflicts we delete most of the rows arriving finally at

R | O | T | I | E | W | N | U | V |
---|---|---|---|---|---|---|---|---|

0 | 2 | 4 | 5 | 6 | 7 | 3 | (8) | (1) |

0 | 2 | 5 | 4 | 6 | 7 | 8 | (1) | (9) |

This gives the only two solutions

```
3496 -
3210 =
------
286 +
286 =
------
572
9516 -
9280 =
------
236 +
236 =
------
472
```

One can verify these solutions by using the python script presented in the first post of this series.