# How do I solve this unreasonable 6x6 game of Towers without backtracking?

I'm playing Towers by Simon Tatham (rules are in the link). This game shares some solution logic to Sudoku.

I recently moved to the last suggested level of difficulty, 6x6 Unreasonable, but I apparenly miss a tool in my belt to solve that level of difficulty because everytime I try one, I can't finish it without backtracking.

I want to solve the following puzzle (playable here), but without using backtracking (that is guessing a square, then coming back to the previous state if I find any conflict). Is there a solution without backtracking or is backtracking the last move I had to learn to master this game?

               4   2
+---+---+---+---+---+---+
|   |   |   | 2 |   |   | 3
+---+---+---+---+---+---+
2 |   |   |   |   |   |   |
+---+---+---+---+---+---+
2 |   |   |   |   |   |   | 2
+---+---+---+---+---+---+
2 |   |   |   |   |   |   | 1
+---+---+---+---+---+---+
|   |   |   |   |   |   |
+---+---+---+---+---+---+
| 2 |   |   |   |   |   |
+---+---+---+---+---+---+
4       1       3   2


And here is my current status:

My current save file (save as any file, but upload it on the game page):

SAVEFILE:41:Simon Tatham's Portable Puzzle Collection
VERSION :1:1
GAME    :6:Towers
PARAMS  :3:6du
CPARAMS :3:6du
SEED    :15:371209209321014
DESC    :41:///4/2//4//1//3/2//2/2/2///3//2/1//,c2z2e
AUXINFO :74:e55bd588ce7f80ffc06c9b2cdb35783c4b0b4022ca6a0f00b6a8156e7761a0de3f273a075b
NSTATES :3:214
STATEPOS:3:213
MOVE    :1:M
MOVE    :6:R2,5,6
MOVE    :6:R5,3,6
MOVE    :6:R0,3,5
MOVE    :5:D-1,3
MOVE    :4:D6,3
MOVE    :4:D2,6
MOVE    :6:P0,4,1
MOVE    :6:P0,4,2
MOVE    :6:P0,4,6
MOVE    :6:P0,4,5
MOVE    :6:P0,0,2
MOVE    :6:P0,1,2
MOVE    :6:P0,2,2
MOVE    :6:P1,5,2
MOVE    :6:P3,5,2
MOVE    :6:P4,5,2
MOVE    :6:P5,5,2
MOVE    :6:P3,4,2
MOVE    :6:P3,3,2
MOVE    :6:P3,2,2
MOVE    :6:P3,1,2
MOVE    :6:P5,0,2
MOVE    :6:P4,0,2
MOVE    :6:P2,0,2
MOVE    :6:P1,0,2
MOVE    :6:P0,0,2
MOVE    :6:P0,0,2
MOVE    :6:P3,1,5
MOVE    :6:P3,1,6
MOVE    :6:P3,2,6
MOVE    :6:P2,0,6
MOVE    :6:P2,1,6
MOVE    :6:P2,2,6
MOVE    :6:P2,3,6
MOVE    :6:P2,4,6
MOVE    :6:P1,5,6
MOVE    :6:P3,5,6
MOVE    :6:P4,5,6
MOVE    :6:P5,5,6
MOVE    :6:P4,3,6
MOVE    :6:P3,3,6
MOVE    :6:P2,3,6
MOVE    :6:P1,3,6
MOVE    :6:P2,3,6
MOVE    :6:P5,2,6
MOVE    :6:P5,1,6
MOVE    :6:P5,0,6
MOVE    :6:P5,0,5
MOVE    :6:P4,0,6
MOVE    :6:P0,1,6
MOVE    :6:P0,2,6
MOVE    :6:R0,0,6
MOVE    :6:P1,0,6
MOVE    :6:P5,5,1
MOVE    :6:P5,4,6
MOVE    :6:P5,4,5
MOVE    :6:P4,4,6
MOVE    :6:P4,2,5
MOVE    :6:P3,2,5
MOVE    :6:P2,2,5
MOVE    :6:P1,2,5
MOVE    :6:P0,1,5
MOVE    :6:P0,2,5
MOVE    :6:P1,3,5
MOVE    :6:P2,3,5
MOVE    :6:P3,3,5
MOVE    :6:P4,3,5
MOVE    :6:R3,4,6
MOVE    :6:R3,5,5
MOVE    :6:P1,4,6
MOVE    :6:P1,5,5
MOVE    :6:P4,5,5
MOVE    :6:P5,5,5
MOVE    :6:P5,4,5
MOVE    :6:P5,4,5
MOVE    :6:P5,4,4
MOVE    :6:P3,1,4
RESTART :41:///4/2//4//1//3/2//2/2/2///3//2/1//,c2z2e
MOVE    :6:R2,5,5
MOVE    :6:R2,5,6
MOVE    :6:R5,3,0
MOVE    :6:R5,3,6
MOVE    :1:M
MOVE    :6:P0,2,6
MOVE    :6:P5,2,6
MOVE    :6:P1,2,5
MOVE    :6:P2,2,5
MOVE    :6:P3,2,5
MOVE    :6:P4,2,5
MOVE    :6:P0,1,6
MOVE    :6:P0,3,6
MOVE    :6:P1,3,6
MOVE    :6:P2,3,6
MOVE    :6:P3,3,6
MOVE    :6:P4,3,6
MOVE    :6:P5,0,6
MOVE    :6:P5,1,6
MOVE    :6:P5,2,6
MOVE    :6:P5,2,6
MOVE    :6:P5,4,6
MOVE    :6:P5,5,6
MOVE    :6:P5,4,5
MOVE    :6:P5,5,1
MOVE    :6:P5,0,1
MOVE    :6:P5,0,2
MOVE    :6:P5,0,5
MOVE    :6:P4,0,6
MOVE    :6:P4,0,2
MOVE    :6:P3,1,2
MOVE    :6:P3,2,2
MOVE    :6:P3,3,2
MOVE    :6:P3,4,2
MOVE    :6:P3,5,2
MOVE    :6:P2,0,2
MOVE    :6:P1,0,2
MOVE    :6:P1,1,2
MOVE    :6:P1,1,2
MOVE    :6:P0,0,2
MOVE    :6:P0,1,2
MOVE    :6:P0,2,2
MOVE    :6:P0,3,2
MOVE    :6:P0,4,2
MOVE    :6:P1,5,2
MOVE    :6:P3,5,2
MOVE    :6:P4,5,2
MOVE    :6:P3,5,2
MOVE    :6:P5,5,2
MOVE    :6:P1,5,6
MOVE    :6:P3,5,6
MOVE    :6:P4,5,6
MOVE    :6:P4,5,5
MOVE    :6:P4,4,6
MOVE    :6:P0,4,5
MOVE    :6:P0,4,6
MOVE    :6:R0,0,6
MOVE    :6:P1,0,6
MOVE    :6:P2,0,6
MOVE    :4:D6,3
MOVE    :4:D2,6
MOVE    :6:P0,3,1
MOVE    :6:P2,1,6
MOVE    :6:P2,2,6
MOVE    :6:P2,3,6
MOVE    :6:P2,3,6
MOVE    :6:P2,4,6
MOVE    :6:P3,1,5
MOVE    :6:P3,1,6
MOVE    :6:P3,2,6
MOVE    :6:R3,4,6
MOVE    :6:P1,4,6
MOVE    :6:P5,0,1
MOVE    :6:R0,3,5
MOVE    :6:P0,1,5
MOVE    :6:P0,2,5
MOVE    :6:P1,3,5
MOVE    :6:P2,3,5
MOVE    :6:P3,3,5
MOVE    :6:P4,3,5
MOVE    :6:R3,5,5
MOVE    :6:P1,5,5
MOVE    :6:P5,5,5
MOVE    :6:P3,1,4
MOVE    :6:P3,2,3
MOVE    :5:D-1,3
MOVE    :6:R5,2,5
MOVE    :4:D6,2
MOVE    :6:P5,1,5
MOVE    :6:P0,4,1
MOVE    :6:P5,4,4
MOVE    :4:D0,6
MOVE    :6:P3,2,3
MOVE    :6:P3,3,3
MOVE    :5:D3,-1
MOVE    :6:P1,1,5
MOVE    :6:P1,1,4
MOVE    :6:P1,2,4
MOVE    :6:P4,1,5
MOVE    :6:R2,1,5
MOVE    :6:P2,0,5
MOVE    :6:P2,4,5
MOVE    :6:R1,1,6
MOVE    :6:P1,2,6
MOVE    :6:P4,1,6
MOVE    :6:R4,2,6
MOVE    :6:P4,0,1
MOVE    :6:R0,2,4
MOVE    :6:P0,1,4
MOVE    :6:P0,4,4
MOVE    :6:R0,4,3
MOVE    :6:R0,1,1
MOVE    :6:P2,2,4
MOVE    :6:P3,2,4
MOVE    :6:P1,4,3
MOVE    :6:P2,4,3
MOVE    :6:P4,4,3
MOVE    :6:P5,4,3
MOVE    :6:R3,3,4
MOVE    :6:P2,3,4
MOVE    :6:P1,3,4
MOVE    :6:P4,3,4
MOVE    :5:D-1,2
MOVE    :5:D-1,1
MOVE    :6:P3,1,1
MOVE    :6:P4,1,1
MOVE    :6:P5,1,1
MOVE    :6:R3,1,3
MOVE    :6:R3,2,1
MOVE    :6:P2,2,1
MOVE    :6:P1,2,1
MOVE    :6:P4,1,3
MOVE    :6:P5,1,3
MOVE    :6:P4,4,1


• +1 for introducing the puzzle type. Aug 14, 2022 at 14:27

Column 5, Row 1:

cannot be 4, as that would violate the clue (3) to the left with 65x24x.

Also:

cannot be 3, as that would create a conflict with
[3,4]
[2,4]
in the top right corner.

Therefore:

must be 5. Continue from there by solving column 5, then the rest of the grid is simple.

Hint for continuation:

Column 5, Row 5 cannot be 2

• Cheers! Thanks for the whole walkthrough, even though I only needed the first clue to finish it. Thank you very much :) Aug 14, 2022 at 10:25
• The walkthrough was mainly for the benefit of others, and to make my answer complete. I expected the first clue to be sufficient. Aug 14, 2022 at 10:35
• Doesn't the content of your "Also:" spoiler block constitute backtracking? Aug 14, 2022 at 17:34
• @Joseph Backtracking involves making logical deductions, based on the assumption that a guess is correct, that may or may not lead to a solution. In this case, we know that the choice is incorrect because there is an immediate conflict. Aug 14, 2022 at 21:11
• Here's the shortest path I see to arriving at a conflict: rot13(Vs pbyhza 5, ebj 1 vf 3, gura pbyhza 5, ebj 2 zhfg or 2, fvapr vg orvat 4 jbhyq ivbyngr gur 2 pyhr. Gura pbyhza 6, ebj 2 zhfg or 4 fvapr jr whfg hfrq hc 2 va gung ebj. Gura pbyhza 6, ebj 1 zhfg or 1 orpnhfr jr whfg hfrq hc 4 va gung pbyhza naq 3 va gung ebj. Ohg abj lbh unir na hanibvqnoyr pbasyvpg: lbh'er tbvat gb ivbyngr gur 3 pyhr ab znggre ubj lbh svavfu gur gbc ebj.) Is there a shorter path to one? If not, then I guess my quibble is that I think that's too long to call "immediate". Aug 14, 2022 at 21:42

Notwithstanding the currently accepted answer (kudos to @Daniel Mathias), backtracking is par for the course at Unreasonable difficulty (hence the name, I suppose), as explained in the full help text:

At Unreasonable level, some backtracking will be required, but the solution should still be unique.

• Oh well, if backtracking is fair game, I'll just keep playing at the previous level. Thank you for pointing this out. I've been playing this game for years so indeed, I didn't check the help when I recently moved up a difficulty level. Aug 14, 2022 at 22:48
• This is interesting to me. For many of these type of logical deduction puzzles, I feel that some hard-level puzzles require 2-3 steps of decisions, or even more, that you have to make "in your head" before you come upon a conflict. So backtracking is just using the crutch of writing out the steps and then "erasing" them when you get to the conflict. It cannot possibly be the case that they are all designed to produce a conflict after only 1 step of decision, can it? Aug 15, 2022 at 15:13
• @qdread sure, why not? Simon Tatham has an interesting guide on how to program these things. Or maybe I misunderstand you? Aug 15, 2022 at 18:08