Pinpoint

Sudoku-style elimination

When solving Skyscrapers, the skyscraper visibility rules of Skyscrapers usually take centre stage. The Sudoku eliminations become sort of a ‘background’ thing that you just carry out on autopilot.¹

But that doesn’t make them unimportant! Sudoku-style deductions carry a lot of weight, and knowing them in-and-out is really beneficial.

I’m sure you’re familiar with the basic rule of Sudoku: each lane (of 9 cells) must contain all the digits $\{123456789\}$ exactly once each, in any order. For Skyscrapers, this means in an NxN Skyscrapers, each lane must contain the $\{1...N\}$ -skyscrapers once each.

Simple rule, but many ways to apply it. Let’s take a grid of unsolved cells:



12345	12345	12345	12345	12345
12345	12345	12345	12345	12345
12345	12345	12345	12345	12345
12345	12345	12345	12345	12345
12345	12345	12345	12345	12345

Suppose we end up solving some cell to be a $5$ -skyscraper:



12345	12345	12345	12345	12345
12345	12345	12345	12345	12345
12345	12345	12345	12345	12345
12345	12345	12345	5	12345
12345	12345	12345	12345	12345

Because a lane can only contain one $5$ , this eliminates $5$ as a candidate from this cell’s row and column:



12345	12345	12345	*1234*	12345
12345	12345	12345	*1234*	12345
12345	12345	12345	*1234*	12345
*1234*	*1234*	*1234*	5	*1234*
12345	12345	12345	*1234*	12345

Suppose we keep going and manage to deduce 3 more lane peaks (pencilmarks ommitted for clarity):



5
			5

		5
	5

There’s precisely five $5$ -skyscrapers in a 5x5 puzzle, so we have one more to find. If we put the pencilmarks in the middle column back, what do we notice?



5		1234
		1234		5
		*12345*
		1234	5
	5	1234

Only the middle cell contains $5$ as a candidate. That’s because every row contains a $5$ , except the middle row.

In this situation, we can pinpoint that the final $5$ -skyscraper must go in the middle cell. There’s nowhere else it could go.



5
				5
		5
			5
	5

If you looked at every skyscraper in a solved puzzle in isolation, it’d have this staggered pattern!

If you’ve solved any Sudoku or Skyscrapers puzzles before, this will probably already be familiar to you. But you might underestimate how often it pops up, especially in comparison to plain Sudoku.

Because of all the added logical deduction, we can very easily end up with very few candidates in a lane. Take this situation:


		2
4	5


		4

Applying Slide to the $4$ -clue half-lane, we get a sequence:


					2
4	12	23	34	5


					4

Then for the upper-right cell, using Sudoku-style elimination we get $[123]$ as candidates:


					2
4	12	23	34	5	*123*


					4

But now look carefully at the top row.

Among all the cells, only one contains $4$ as a candidate. This means we can pinpoint $4$ to that cell. Bingo!


					2
4	12	23	4	5	123


					4

Very simple, but powerful deduction. Always be on the watch out for this, because it can sometimes sneak by you, especially in larger puzzles!


					2

	5
2	34	2	13	6		345

					6

Can you spot a deduction here?

This example also happens to illustrate why pencilmarking can be so essential. What happens if we pencilmark the remaining cell here? (using Leap of Faith)


					2
					45
	5
2	34	2	13	6	*134*	345

					6

Once we have pencilmarks in all cells of a row, we might notice there’s a skyscraper we can pinpoint which we didn’t see before…


					2
					45
	5
2	34	2	13	6	134	5

					6

If we hadn’t pencilmarked, we might’ve missed that! If this were an easier puzzle, we’d probably end up reaching that later on anyway, but in a tough puzzle this could be the only deduction we’re able to make, and if we didn’t spot it we’d be stuck.

Personally, I love this setup. For me the rules of Sudoku are too… dry? to warrant a full puzzle, and puzzles that build upon it with an additional constraint (e.g. Futoshiki) are far more fun.↩