The 1-level puzzles normally require greater ingenuity. Some indirect logical inference will be required. For example, you might recognize that a particular letter X in a particular square would make some other puzzle piece or some other row or column not admit that letter X. And by eliminating the possibility of X in that square, there may be a letter you can fill in. By comparison, the computer (as currently configured) can't infer that that other puzzle piece or row or column would lead to a dead-end until it actually makes the "guess" of filling in X. Sometimes, however, the level 1 puzzles get quite annoyingly difficult indeed.

The 2-level puzzles can require quite a bit of chicken-scratching until they fall into place. Often, I've found, whether you can fill in a few letters at the beginning makes a big difference. The puzzles for which the very first letter (the seventh letter, that is!) is hard to find seem to be very difficult indeed.

In particular, the reasoning I find most difficult is simultaneous reasoning about more than one letter at a time. I try to determine whether for some letter X I can find six places to put X that are all legal, taking into account only X and the letters already filled in. Another approach is to find pairs or triples of squares that, regardless of what letter fills them, must be the same. This helps eliminate a wider range of possibilities.

I mostly don't try the higher-level puzzles, though the occasional unexpected success keeps me going. Remember that the computer's sense of logic is limited, and that the human's logical inference (and sense of instinct about what to focus on) can leap many levels. The highest-level puzzle I know of that was successfully solved (not by me!) is seven.

Curiously, there's no real room for "heuristics" in this analysis -- if by heuristic one means a way of gauging which ambiguous square should be filled in with a guess. There are two reasons for this. One is the observation that the solution process often looks for proofs that some cell prospect leads to an invalid dead-end, rather than trying to find a shortcut to an example solution. Another factor making heuristics problematic is the uniqueness property for these puzzles. That is, the puzzles are pre-chosen from among those that have only one solution. A heuristic useful in other contexts is "preserving options", here choosing the square that leaves the largest number of alternatives still possible. But this will likely be a poor strategy, because we know in advance that there is only one solution to the puzzle. Instead, following the square that seems to offer a narrow chain of logical implications each requiring only one solution) may be more likely to uncover the unique solution promised by the algorithm.

That said, the advertised "difficulty level" is an objectively evaluated measure, based on the computer-generated solution. Simply put, the difficulty level is the number of "guesses" that a reasonably intelligent automaton would take before encountering the solution. This automaton can fill in any cell that can have but one possible letter, and can insert any letter that can have only one possible position (among the six in a given line or piece). Then those conclusions can If the solution can be found deterministically, by filling in one "must" after another until the entire puzzle is complete, then it is described as level "0". Otherwise, the difficulty level is the number of guesses or branch points (technically, the recursion depth) encountered until the solution is obtained.[1]

Perhaps a better measure of difficulty is the number of branch points encountered until all other possibilities are proved wrong. For difficulty zero, the two measures are the same.

A guarantee.

So far, though, I haven't found any example of reasoning that takes advantage of knowing that the solution must be unique. This reasoning step is certainly possible -- for example, a 2x2 square of AB and BA automatically admits a transposition of both[3], hence a duplicate solution, hence is invalid -- but the reasoning has never been material to finding a particular 6x6 solution. If you happen across such a case, do send me mail!.

Special rules.

The puzzles on this page reflect an additional constraint. We toss out tilings including the 2x3 rectangle, because those puzzles seem relatively boring (and there are so many others to choose from!).

This still leaves lots of tilings. Starting with a blank tiling, the algorithm constructs all possible legal completions. (Without loss of generality, but precisely, we can assume A-F across the top row.) Then, for each possible collection of six distinct letters anywhere in the grid, find out how many legal completions match those six letters. If exactly one, a puzzle setting has been identified. For cosmetic reasons, we apply a random rotation-and-reflection, and then reassign the A-F letters in accordance with the alphabetical left-to-right, top-to-bottom convention of the published article. Typically, there are anywhere from several dozen to several hundred legal completions for a given tiling, leading to between fifty thousand and two hundred thousand correct settings. Overall, there are approximately half a billion total puzzles (before rotation and reflection) that satisfy the desired conditions. You won't solve them all! (You probably won't even ever solve as many as one for each tiling.)

The corresponding quantities for the 4x4 puzzle and 5x5 puzzle are two thousand[3] and two hundred thousand. The series of numbers 2,000; 200,000; and 500,000,000 suggests that that the number of possible puzzles of size 7x7 is humongous, so the exhaustive enumeration of those puzzles is unlikely.

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