First in a series with puzzle solving tips. This time, TomTom solving tips for last week’s hard Thursday puzzle.

People always ask me how to solve puzzles and fast. My first answer is practice, practice, practice! My second is notation, notation, notation! After that, I say learn how to construct interesting puzzles and run through more involved logical deductions than most puzzles tend to have. In this column I intend to dissect some of our Grandmaster Puzzles with such interesting properties and at the same time reveal ways I look at puzzle styles that might help out in the future. This week’s topic is the Thursday TomTom, advertised at medium-hard difficulty, that few solvers beat in an expected time. While it shouldn’t need saying, SPOILERS AHEAD!

“Basic” TomTom global steps, specifically that the total sum is 21 for all rows in a 6×6 puzzle, should get the solver to this position. Also, while this is a NoOp puzzle, all the 13’s are clearly + cages.

A similar early observation is to see that all two-cell 1 cages are subtractions (try to think of a counter-example). In fact, this NoOp puzzle is turned effectively into an Op puzzle very quickly. Only three 3 cages have any flexibility, the top one is resolved quickly, and the other two can be resolved much later in the solve.

The key to this puzzle is to consider the unusual grouping of four 1- cages on the left side collectively.

The 4 left cell entries in the subtraction cages are 6, 5, 4, and 2. Each of these entries needs a partner in column 2, one digit away. So these can be 5, 4 or 6, 3 or 5, and 1 or 3 respectively. But if 6 becomes 5, then 4 cannot also become 5 in the second column. 4 becomes 3, and from similar logic 2 becomes 1. The 5 can be either a 4 or a 6 (for logic I will consider both options). But uniqueness arguments force 5 to go to 4. If not, then a 65 and 56 which are completely interchangeable exist in the left of the puzzle giving 2 solutions.

An alternate way to see this without diagramming all the sets is to consider parity. If the two numbers not in the four 1- cages in the leftmost column are odd (1 and 3), the two numbers not in the four 1- cages in the second column are even. In the 25 at the top, the 2 must be to the left.

(Aside: “Uniqueness” — making a deduction because the consequence of any other possibility is to have more than one solution — is a useful concept for speed solving but typically frowned on from a standpoint of mathematical rigor because such steps do not conclusively prove there is just one answer. I’d say they “feel bad” as a solver, almost like cheating. Not cheating in the “Eugene” sense, but maybe cheating oneself out of a better experience. Still, sometimes discovering them is an unexpected Aha moment and that is what many people commented on with this puzzle, getting a 6 in R6C2 much sooner by doing so. Why did they feel comfortable using uniqueness here? Well, I know how to write valid puzzles. Grandmaster Puzzles only have 1 answer (free t-shirt to anyone who first reports a mistake on this front), so this is a special and trusted puzzle setting where uniqueness probably feels even worse to use but will always get the right job done quicker.)

Identifying the sets in the first two columns places a 2 in R1C2, and a 4 or 6 in R6C2. Critically, it also puts a 5 into column 3 at the top. Now, the key row is the second from the bottom. Where can the 5 go in this row? If it is in columns 1 or 2, it drags in a 4 and a 6 in one way or another (one is in the other column 1/2 cell, the other in column 6). That leaves 2 and 3 in the three-cell 13+ cage so cannot work. Instead there must be a 5 in R5C4 and the puzzle solves in a more standard way from there. Note that uniqueness is not needed at all; quickly after placing the 5 in R5C4 you get a 4 in R6C3 that resolves the 4/6 question.

I hope this is a revealing look at what is a quite unusual TomTom. While our puzzles may get hard, there is always an interesting logical route and the discovery process will be worth persevering through. So use uniqueness, (or even worse “guess”) at the risk of compromising your own enjoyment! Some get joy from solving puzzles very fast. Others from solving very good puzzles. Some from both. Choose your own path through these Grandmaster Puzzles.