The Stack

After dithering with Cryptics for a while, I finally assailed Cross Numbers 1. It turned out that it contains only nine obviously nonsensical clues. I gave myself permission to use the Hint button without limit on those, but on my first pass with that rule, so great was my habitual reluctance to press that button it that I wound up only using it on one number. It turns out the clues contain a lot more information than is strictly necessary, so having nine clues that don’t really provide any information isn’t as big a handicap as you might imagine.

In particular, the ways that the clues reference each other means that you can often apply them in reverse. A clue like “34-Across: Sum of 9-Down and 12-Across” might as well be “9-Down: Difference between 34-Across and 12-Across” or “12-Down: Difference between 34-Across and 9-Down”. Really, the arrangement of the clues, the way that there’s a clue for each number in the grid, is a lie. You don’t necessarily figure out what’s in 34-Across from the clue labeled “34-Across”, and you’re not necessarily finished with that clue just because you’ve filled 34-Across into the grid.

Still, in my first pass I wound up with an irreconcilable logical contradiction. I restarted, this time taking hints for all the nonsense clues. I wound up with the same contradiction, but this time kept on going, ultimately getting a rating of 15%, even though the grid was mainly filled in correctly, just because of all the hint penalties I had incurred. Afterwards, I checked the game’s answers against the clues, and found that, as I suspected, my recurring contradiction was the result of a non-obvious mistake in the clues — one where the referenced numbers all exist and have a reasonable number of digits, but the math doesn’t work out as promised in the official solution.

This raises an interesting question: Given that there are multiple clues containing mistakes, how can I have any confidence in any of them? The answer is that correct things tend to confirm each other. If I have a number that already has a few digits filled in from the numbers crossing it, and I do some arithmetic to find that number and get an answer that fits the digits in place, that tends to convince me that the clues for both that number and the numbers crossing it are valid. Even better: If I notice a contradiction, and check my math, and find a mistake that resolves the contradiction, that really gives me trust in those clues, because it shows that I can trust them more than I can trust my own thoughts. Because of this, I had a pretty good idea of which of the seemingly contradictory clues would turn out to be wrong: it would be the one with the least confirmation. This turned out to be the case.

It strikes me that one could make a puzzle out of this, similar to how diagramless crosswords were, according to legend, originally inspired by an incident where a crossword’s clues were accidentally delivered for publication without a grid. The challenge would be to find the deliberate mistakes in the clues, given that most clues are error-free. To make it easier, you could let the solver know the exact number of mistakes — although if you did that, you could reject solutions that don’t have enough mistakes!

Anyway, I’ve tried to figure out what the faulty clues were supposed to be, to see just how badly things went wrong here. It turns out that most of them are pretty close to something reasonable. The most common error, apart from leaving off the second term of a sum, is just mixing up Across and Down.
Here’s my list:

• 15-Across: Sum of 3-Across and 34-Down: Should be 30-Across and 34-Across.
• 17-Across: Multiple of 45-Down: Should be 45-Across
• 21-Across: Sum of 46-Across: …and 32-Down
• 46-Across: Average of 35-Across and 57-Across: 57-Across should be 57-Down
• 61-Across: Sum of 47-Across: …and 50-Across
• 69-Across: Sum of 16-Across: …and 9-Down
• 9-Down: Square of 71-Down: Should be 71-Across
• 24-Down: Product of 24-Across, 47-Across, and 68-Down: 24 Across should be 25-Across
• 40-Down: Product of 48-Across and 70-Across: Should be 58-Across and 70-Down.
• Finally, the stealth mistake. 16-Across: Sum of 46-Across, 61-Across, and 42-Down: The best I’ve come up with for this is replacing 46-Across with 25-Across. But that’s a relatively unlikely typo, so I’m not really satisfied with this solution. Maybe there’s a better one that has typos in two of the terms.