How to play
The box hides exactly K atoms, one per cell, never two in a cell. The number K is given; their positions are not.
A ray has been fired into the box from every port on the ring, and each port carries the result. Those results are the clues — the whole table is given, the way another puzzle gives you its edge numbers. A ray travels in a straight line into the box until it interacts with an atom.
Hit (H): if the next cell along the ray’s path holds an atom, the ray is absorbed and never comes out. The port shows H.
Deflection: if the next cell is empty but the cell diagonally ahead-left of the ray holds an atom, the ray turns 90° to its right; an atom diagonally ahead-right turns it 90° to its left. The ray does not advance while it turns, so it may turn several times in a row and may leave from any port on any side.
Reflection (R): if atoms sit diagonally ahead on both sides, the ray reverses and comes back out of the port it entered. A ray also reflects when the atom sits diagonally beside the entry port itself — it never enters the box at all — or when a chain of deflections happens to send it back out of its own port. The port shows R.
Detour: any ray that comes out of a different port is a detour. Its entry port and exit port both show the same number. Detours are two-way: each number appears at exactly two ports, which are the two ends of one ray.
Mark where you believe the atoms are. The puzzle is solved when you have placed exactly K atoms and your placement reproduces every result on the ring.
Every published puzzle has exactly one placement of K atoms that reproduces the full ray table, so the correct answer is never a guess between equals.
A straight-through detour clears three lines
When a ray goes in one side and comes straight out of the port directly opposite, the simplest explanation is that nothing touched it — and that clears far more than the line it travelled. Every cell on that line is empty, and so is every cell diagonally beside it, because an atom there would have bent the ray away.
Example: The top of column 4 and the bottom of column 4 carry the same detour number. Rule out the whole of column 4, and columns 3 and 5 as well.
Open with the corners
A corner port has only one diagonal neighbour inside the box instead of two, so it can only be deflected one way. That makes its result the least ambiguous on the board, and a corner straight-through strikes two whole lines off at once.
Example: The leftmost top port and the topmost left port both come straight through. That is two lines and their neighbours ruled out before you have marked a thing.
Read an R as a statement about two cells
An R most often means the ray never got in: the atom sits diagonally beside the port, in one of the two cells flanking the entry cell. It is a strong clue precisely because it names such a small area — and if both those cells turn out to be empty, the reflection happened further in and the ray turned back on itself.
Example: The left port of row 3 reads R. Either the cell at row 2 column 1 or the cell at row 4 column 1 holds an atom — and if you have already cleared one of them, the other is certain.
An H is a cone, not a point
A hit tells you the ray met an atom head-on, but not where. Walk the ray from its port through the cells you have already cleared: the first cell you cannot rule out is the earliest candidate, and each further cell along the path is another. The value of an H rises as the cells before it are eliminated.
Example: The top of column 6 reads H and you have already cleared rows 1 and 2 of that column. The atom sits somewhere from row 3 down — and one more cleared row shortens the list again.
Test an atom against a port you have not used
Once you are confident about one atom, trace what a nearby ray would do with it and compare that against what its port actually says. If the two agree, the atom is confirmed and so is everything along the path; if they disagree, you have found your error while it is still cheap to fix.
Example: You believe there is an atom at row 2 column 5. A ray from the left of row 1 should then bend down and leave at the bottom of column 4 — check whether those two ports really do share a number.
Two atoms can send a ray back the way it came
A ray does not advance while it turns, so a second atom can catch it mid-turn and bend it again. Two atoms placed a knight’s move apart can turn a ray twice and hand it straight back to its own port, which is why an R does not always mean an atom beside the entry.
Example: A port reads R but both cells flanking its entry cell are already ruled out. Look for a pair of atoms further along the line that could have turned the ray twice.
Ruled-out dots are the real work
Almost every deduction here is negative: a result tells you where atoms are not far more often than where they are. Dot every cell a result eliminates as you work around the ring — the atoms end up being the cells nothing ever ruled out.
Example: After reading half a dozen straight-through detours, one column has a single undotted cell left and the tally still needs an atom. That cell is one.
Check the ports you never read
The board only accepts a layout that matches every port on the ring, so a layout built from a handful of easy clues can still fail on one you skipped. When your K atoms are placed and the board has not solved, the ports drawn in red are the ones your atoms disagree with — press one to see the path your own placement implies.
Example: All five atoms are placed but the puzzle has not solved, and one port on the far side is red. Press it and follow the ray your atoms send out.