How to Solve Stargazer

The rules of Stargazer

A Stargazer grid is mostly empty, with a scattering of arrow cells. Each arrow points in one of eight directions, the four straight ones and the four diagonals. Your job is to place stars in the empty cells so that four rules hold at once.

First, an arrow cell never holds a star. Second, each arrow points at exactly one star. An arrow's ray runs from its own cell in the arrow's direction all the way to the edge of the grid, and exactly one cell on that ray holds a star. Third, each star is pointed at by exactly one arrow, so no star sits on two rays and no star sits off every ray. Fourth, the numbers outside the grid are exact star counts for that row or column, and every row and every column carries a number at all three difficulties.

There is no adjacency rule in Stargazer. Stars may sit side by side or corner to corner, which is a real difference from Star Battle and Star Circle. A ray also passes straight over any arrow cell in its path and keeps going, so nothing ever blocks an arrow's sight.

Two invariants worth memorising

The number of stars equals the number of arrows. Count the arrow to star pairings from the arrow side and you get one per arrow, count them from the star side and you get one per star, so the two totals must match. That gives you the board's total star count before you place anything, and it must equal the sum of the row numbers and the sum of the column numbers.

The second invariant does most of the work. Call a cell's cover the number of rays passing through it. A cell with cover zero can never hold a star, because no arrow would be pointing at it. A cell with cover two or more can never hold a star either, because two arrows would be pointing at the same star. Only cells with cover exactly one are candidates, and each candidate belongs to exactly one arrow. The whole puzzle reduces to picking one candidate for each arrow so that the row and column counts come out right.

The opening move

Start by working out the cover of every cell. Tap an arrow to light up the line it points along, then read off the cells it touches. Tap it again to clear the trace. Tracing is a viewing aid, it submits no move and changes nothing on the board.

Trace every arrow in turn and mark two kinds of cell as excluded. Cells that no ray reaches are out, they fail the one arrow per star rule. Cells that two or more rays reach are also out, for the same rule read the other way. On a typical hard grid this clears a large part of the board before you have placed a single star, and what remains is one private set of candidate cells for each arrow.

Read the counts against what is left

Now compare each row and column number with the candidates that survived. If a line's number equals the number of candidates left in it, every one of them is a star. If a line's number is zero, every candidate in it is excluded, which often wipes out candidates for arrows pointing elsewhere.

The reverse check is just as useful. Once a line has as many stars as its number allows, every remaining candidate in that line is excluded. Alternate between the two readings, because each exclusion shrinks a candidate set somewhere else and can trigger the next deduction.

Arrows with one candidate, lines that are full

If an arrow has exactly one candidate left, that cell is a star. Place it, then apply the consequences at once. The star fills a slot in its row and its column, and every other candidate in a line that has now hit its number is excluded.

This pair of rules, single candidate for an arrow and full line for a count, will finish most easy boards on its own. Run them to exhaustion before reaching for anything harder, because each new star usually unlocks two or three more.

Arrows confined to one line

Look for an arrow whose surviving candidates all sit in the same row, or all in the same column. That arrow must spend its star inside that line, whichever candidate it eventually takes. You now know one of the line's stars is committed even though you cannot say which cell holds it.

The counting argument follows. If a row's number is two and two different arrows are both confined to that row, those two arrows account for the whole row, so every candidate in the row belonging to any other arrow is excluded. A row whose number is one with two arrows confined to it is a contradiction, which is how you refute a branch.

Confinement across several arrows and lines

The general form is a matching argument. Take a set of arrows and count the distinct lines their candidates touch. If a set of arrows can only be satisfied inside a group of lines whose numbers add up to exactly the size of the set, those lines are fully spoken for, and every candidate in them belonging to an arrow outside the set is excluded.

This is the technique that separates hard boards from medium ones. Work it in small sets first, two or three arrows, and only widen the net when nothing smaller moves. On Puzzle Lair every Stargazer puzzle is guaranteed to have exactly one solution reachable by logic alone, so if you are stuck there is always a deduction waiting, never a guess.

Work through a small example

Picture a 6x6 board with an arrow in the top left cell pointing east. Its ray is the rest of the top row, five cells. Suppose the top row's number is one. That single star must be one of those five, which is consistent, so the row number tells you nothing yet on its own.

Now add a second arrow lower down pointing north east, whose ray crosses the top row in the fourth column, and suppose a third arrow's ray also crosses that same cell. The fourth column cell has cover two, so it is excluded, and the east arrow drops to four candidates. If the column numbers then rule out two more of them, the east arrow has two candidates left, both in the top row, so it is confined to the top row. Since the top row's number is one, that one star belongs to the east arrow, and every other arrow's candidate in the top row is excluded. Each exclusion feeds the next line count, and the chain carries the board to its single solution.

Marking, checking, and finishing

Tap a cell to cycle it blank, star, excluded, and back to blank. Excluded is an optional solving note, a way to record a cell you have ruled out, and blank and excluded both count as empty when the board is graded. Use the excluded mark freely during the opening cover pass, it is the fastest way to keep track of what the rays have already ruled out.

Nothing is flagged while you solve. When you think the board is finished, press Check and it tests the whole grid at once. A failed Check reveals no cells, costs no mistakes, and never ends the attempt, so you are free to experiment. Stargazer is currently marked beta, so rules and tuning may still change as the puzzle is tested.