Laser mirror puzzles, and how Fibs’ Laser Chase bends the beam

The one rule: angle in, angle out

Every laser puzzle runs on the law of reflection: a ray leaves a mirror at the same angle it arrived. It’s in Euclid’s Optics, and in the first century Hero of Alexandria showed something better: the rule follows from a simpler idea, that a reflected ray takes the shortest path it can.

You can see why with a trick that still feels like cheating. Say a ray goes from A, off a mirror, to B. Flip B to the far side of the mirror, the way a reflection sits behind the glass. The shortest route from A to that flipped B is a straight line, and the spot where the line crosses the mirror is exactly where the real ray bounces. A bounce is a straight line in a folded world.

Puzzle makers keep only the tidiest case. Stand a mirror at 45 degrees across a square. A beam meets it at 45 degrees and leaves at 45 degrees on the other side, so it turns by exactly 90 degrees. That gives you two mirrors and four ways in:

• Leaning right, like a slash. A beam heading right goes up, and one heading left goes down. Heading up, it goes right, and heading down, it goes left.
• Leaning left, like a backslash. A beam heading right goes down, and one heading left goes up. Heading up, it goes left, and heading down, it goes right.

Whichever way a beam arrives, one setting turns it left and the other turns it right. So on a grid, a mirror is really a yes-or-no question, and a laser puzzle is a chain of them.

Why a few mirrors make a good puzzle

Each mirror has two settings, so every mirror you add doubles the possibilities. A weekday room in Fibs’ Laser Chase has five to seven mirrors, which is 32 to 128 settings. Sunday’s has eight or nine, which is 256 or 512. You’d never try them all, and you never need to, because a beam only asks about the mirrors it actually reaches.

Turn a mirror the beam doesn’t touch and nothing happens. So the puzzle isn’t really a list of settings. It’s a tree: the beam reaches a mirror and splits into two possible futures, each runs on to its next mirror or dies on a wall, and most branches die quickly. The small room below has 16 settings but only 5 different beams, and just 1 of those lands on the cat.

That’s also where a good room gets its bite. A wrong turn that hits a wall at once costs you nothing, since you can see it fail. A wrong turn that bounces on off three more mirrors before it dies can keep you busy for a while. When our program grades a room, those are the things it counts: how many different beams the room allows, and how far the wrong ones run.

A beam is an unusually honest puzzle piece, too. Everything the game knows is drawn in front of you. Turn one mirror and everything after it changes at once, while everything before it stays put, so you always know which mirror to look at next.

How to solve a laser mirror puzzle

Here’s a small room, drawn for this post and not from any daily. The laser shines in from the left wall. Furniture stops the beam, and so do the walls. Each of the four mirrors can stand either way, and you want the dot on the cat.

1. Start at the target, not the laser. A beam can only reach the cat along her row or her column. There’s no mirror in her column and just one in her row, up in the top right. So that mirror has to send the beam left.
2. Ask what feeds that mirror. To leave heading left along the top row, the beam must arrive from below, running up that column. There’s one mirror down there, so it has to send the beam up. You now know the last two mirrors without having tried a thing.
3. Now go forward and meet it. From the laser, the beam reaches the first mirror. Up runs straight into the top wall, so it has to go down. That reaches the second mirror, where left runs into the plant, so it has to go right. And right is the mirror you were aiming for.

Four mirrors, no trial and error, and the room started just two taps away. Backward is nearly always quicker than forward, because the target has so few ways in.

Six habits that crack most rooms

• Work backward. List the squares that can send a beam to the target. There are usually very few.
• Mirrors only talk along lines. A mirror can only feed another in the same row or column, with nothing in between. Most pairs of mirrors can’t talk at all.
• Count the bounces. Every bounce swaps the beam between running across and running up or down. It starts running across, so after an even number of bounces it’s running across again, and after an odd number it’s running up or down. In Fibs’ Laser Chase the answer uses each mirror once, so count the mirrors: an even count means the last leg runs along Fibs’ row, and an odd count means along her column.
• No mirror is spare. A room with exactly one answer can’t have an unused mirror, because that mirror could stand either way, which would make two answers. If your route skips a mirror, it isn’t the route.
• Read the furniture. A piece in an odd spot is often blocking a route that would otherwise also work. Ask what it’s there to stop.
• Fix the beam from the laser outward. A mirror the beam never reaches can’t be the problem yet, so don’t touch it until the beam arrives.

Mirrors have stranger corners than you’d think

The room a light can’t fill

Here’s a question that sounds easy. If every wall of a room is a mirror, will one candle light the whole room? Ernst Straus asked it in the 1950s, and this is the illumination problem. In 1958 Roger Penrose showed the answer can be no, with a curved room built from ellipses that always leaves dark patches. Straight walls took much longer. In 1995 George Tokarsky built a 26-sided room with a spot that a candle at a certain other spot can never light, however many times the light bounces. Then in 2016, Samuel Lelièvre, Thierry Monteil and Barak Weiss showed how small such gaps are in a large family of straight-walled rooms: a light leaves out at most a finite number of points. Our rooms are much friendlier, since there’s always a way to the cat.

The mirrors on the Moon

Set three flat mirrors at right angles, like the inside corner of a box, and you have a retroreflector: any ray that goes in comes back the way it came. Apollo 11, 14 and 15 left arrays of them on the Moon. With two Soviet rovers and India’s Chandrayaan-3 lander, that makes six reflectors in all, and scientists on Earth still aim lasers at them to time the round trip. It’s a hard shot: of roughly 3 × 10¹⁷ photons in a pulse aimed at a reflector, only about 1 to 5 come back. That’s how we know the Moon is drifting away by about 3.8 centimeters (1.5 inches) a year.

A short history of laser puzzles

1977: Black Box. Eric Solomon’s Black Box turned the idea inside out. One player hides atoms in a grid, and the other fires rays in from the edge, using where they come out to work out what’s inside. Waddingtons published it, and Parker Brothers followed. Acornsoft later put it on the BBC Micro.

1987: Deflektor and Laser Chess. Costa Panayi of Vortex Software made Deflektor, published by Gremlin Graphics, inspired by scientific software that worked with lasers. You rotate mirrors to deflect a beam and destroy all the cells on each level, and you can overload the system if you’re careless, which makes it the same idea as ours with higher stakes. The same year, Mike Duppong’s Laser Chess won a programming contest run by Compute!’s Atari ST Disk & Magazine: a two-player game where a shot bounces off any mirrored piece.

2005: Khet. Michael Larson, a professor at Tulane University, designed a game with his students Luke Hooper and Del Segura as a class project. It shipped as Deflexion, with real low-power lasers, and after a trademark dispute it became Khet, with an Egyptian theme. You move a piece, fire, and try to bounce the beam into the other side’s Pharaoh.

Pencil-puzzle fans got their own version when David Millar invented the Haunted Mirror Maze, in which the mirrors bend your line of sight and the monsters depend on how you look at them. It’s the same geometry with a sense of humor.

Where to play laser and mirror puzzles

We played these three in a browser before writing about them. All are free.

• Black Box. In Simon Tatham’s free collection, and the most surprising of the three, because you never see the beam. You click the edge of a dark box to fire. An H means it hit a ball, an R means it came back, and a matching pair of numbers shows where a beam went in and came out. We found three balls in an 8 × 8 box by working from those marks alone, and once all three were marked a button appeared to check them. Its manual explains how a ball near a beam bends it.
• Undead. Also Tatham’s, and this is the Haunted Mirror Maze: fill a grid of diagonal mirrors with ghosts, vampires and zombies, with numbers around the edge counting who can be seen along each line. The rule that makes it is that vampires can’t be seen in a mirror and ghosts can only be seen in one. The 4 × 4 we solved had nine monsters to place, and tracing those bent sight lines is good practice for any mirror puzzle.
• Pathway. A calm, pretty indie game and the closest here to ours: a beam, a 5 × 5 board, and pieces of glass you tap to rotate until the light reaches a bulb. Each level has its own name and color, and a line of advice. We played the first two levels, which are gentle, and level three asks you to drop the beam, fold it left and lift it into the bulb. Its page has more.

If you like bent lines and sight lines, you may also enjoy the logic puzzles in games like Sudoku.

Fibs’ Laser Chase: a daily mirror puzzle with a cat in it

Fibs’ Laser Chase is our free daily laser puzzle. The Kid shines a laser pointer into the living room, and Fibs the cat wants that red dot. The laser comes in from the left wall, furniture and walls stop the beam, and tapping a mirror flips it the other way. There’s nothing to drag. Land the dot on Fibs and she pounces.

A new room comes every day at midnight, your time, the same one for everyone. Monday to Saturday it’s 7 × 7 squares with five to seven mirrors, and Sunday’s is 7 across and 9 tall with eight or nine. The Kid has always set some mirrors the wrong way, and the count under the room shows your taps and the fewest it can take. Every room has a par time, about what the fastest third of players take, and Share sends your room’s number, your time and your taps, never how the mirrors stand. Your first room is a tiny practice one with a few guided steps.

How each room is made

Each day’s room is made from the date by a program, so everyone gets the same one. It starts with the answer. The beam leaves the laser and runs a few squares to a mirror, turns, runs on to the next, and so on, until the day’s mirrors are placed and it reaches Fibs. It may cross its own path over bare floor, but never through a mirror. Then furniture goes down off the beam, and the maker looks for any other setting that also reaches her, and puts a piece in its way, until only the answer is left. Last, the Kid turns some mirrors the wrong way. In our tests, a separate counter tries every setting, one by one, to check the maker’s work. How we make puzzles with exactly one answer has more on how the family’s dailies are checked.

Each candidate is graded by the choice tree described above, and the day’s room is the one closest to the day’s target. So when a room feels like a set of tidy decisions rather than a pile of guesses, that’s by design. How to play has every rule, and there’s a new room waiting.

Questions people ask