A Masyu loop cannot cross itself or revisit any cell other than the starting cell used to close the route, because every Masyu board represents one simple orthogonal loop drawn on a grid of unit squares. The line moves only between orthogonally adjacent cells, occupies each body cell at most once, and may only touch the start cell a second time to seal the loop, which means every attempt to reuse a non-starting cell or to draw one segment on top of another is rejected as soon as the browser detects the duplicate step.
Masyu is a Nikoli-style pearl loop puzzle where black and white pearls impose strict local turn and straight-line requirements on the path, but the crossing question is settled before those pearl rules ever apply: a simple loop, by definition, cannot intersect itself, branch, or backtrack, and these structural limits hold on every board size, including the compact four by four version used by Masyu Puzzle.
What "Simple Loop" Means in Masyu
The phrase "simple loop" is the technical term used in graph theory and in puzzle writing for a closed path that visits every vertex at most once, and Masyu inherits that definition directly from its loop-drawing ancestry. In practical terms, a Masyu solution is a sequence of cells where each consecutive pair shares an edge, the first and last cells are the same, and no cell appears twice in the middle of the sequence. This is why a Masyu grid shows a single continuous ribbon rather than the braid of overlapping tracks you might see in other pencil puzzles.
When the puzzle says the route must not "cross or revisit itself," those two restrictions are the same rule viewed from two angles. Crossing means drawing a new segment that would overlap a cell already occupied by an earlier segment, which is also a form of revisiting. Reusing a cell means stepping back into a cell the path already owns, which on a square grid either creates a self-crossing or forces the loop to branch. Both are excluded by the simple-loop rule, so the browser validation only needs one check: did the proposed step land on a cell that is already part of the route, apart from the start cell at closure?
This single check is what makes Masyu noticeably cleaner than loop puzzles that allow crossings. You do not have to plan for "which line is on top," and you do not have to memorize edge-priority rules, because the path is always flat, always non-self-intersecting, and always exactly one cell wide on the four by four keyboard board.
Why Crossing Would Break Masyu
If a Masyu loop crossed itself, the rule for the white pearl could no longer be satisfied consistently. A white pearl demands that the line pass straight through it and that at least one of its immediate neighbors carries a turn, so any cell that the path visits twice would either fail the "straight through" test, fail the "neighboring turn" test, or both, depending on which arm was duplicated. With a black pearl, the rule is even stricter: the path must turn on the pearl and continue straight on both arms, and that turn cannot be expressed on a cell that is already part of another arm of the same loop.
Beyond the pearl constraints, a self-crossing loop also breaks the scoring model used in the keyboard board. The Masyu Puzzle score starts at 1,000 and subtracts ten points for every accepted path step, so a clean twelve-step perimeter solution lands on 880. If the board allowed crossings, two extra accepted steps would only cost twenty points and the score would no longer reward the cleanest solution, which is part of why the engine simply refuses the step that would create the duplicate.
The validation is deliberately conservative. Once the path enters a body cell that is already part of the route, the browser reports a detected dead end and the line cannot be extended further from there, because extending would either branch the loop or push the path back over itself. The clean way to leave that state is to restart from the original cursor position with an empty path.
How to Draw a Valid Masyu Loop
The keyboard workflow on the fixed four by four board is built around the simple-loop rule, so every control assumes a single non-crossing line. The steps below produce the disclosed twelve-step perimeter solution and the matching 880 score.
- Move the cursor to the black pearl in the upper-left corner of the four by four grid; this is the starting cell where the loop will also close.
- Press Space or Enter on the black pearl to begin the loop, which marks that cell as the start of the path and locks the cursor there.
- Press the right arrow key three times so the path travels across the top row of the grid, passing straight through the white pearl that sits immediately to the right of the black pearl; each accepted arrow step subtracts ten points from the starting 1,000-point score.
- Press the down arrow key three times so the path travels down the right edge of the grid toward the bottom-right corner.
- Press the left arrow key three times so the path travels back across the bottom row of the grid toward the lower-left corner.
- Press the up arrow key three times so the line climbs the left edge of the grid; the third up arrow arrives at the black pearl and closes the route at the start cell.
- The browser runs the closure test, checking for unique body cells, clue coverage, straight passage with a neighboring turn at the white pearl, and a turn with two straight arms at the black pearl.
- If every check passes, the closed route is reported as solved and the final score is displayed; the disclosed clean solution is 1,000 minus ten times twelve accepted steps, which equals 880.
- Press the Restart control to restore the original cursor, an empty path, the starting 1,000-point score, and the unsolved status, so you can replay the route without reloading the page.
Pearl Rules That Rule Out Clever Crossings
The pearl constraints give the simple-loop rule its Masyu-specific flavor, because they turn what could have been a relaxed drawing puzzle into a tight local-logic exercise. On a white pearl, the line must pass straight through the cell and turn in at least one of its immediate neighbors, which is why the disclosed solution runs straight across the top edge and turns at the top-right corner instead of dipping through the white pearl at an angle.
| Pearl Type | Path Behavior On the Pearl | Required Neighbor Behavior | Effect If Violated |
|---|---|---|---|
| Black pearl | Line must turn on the pearl itself | Both adjacent cells stay straight | Path is rejected at closure |
| White pearl | Line must pass straight through the pearl | At least one adjacent cell carries a turn | Closure check fails |
| Empty body cell | Any straight passage is allowed | No local constraint | None on its own |
On a black pearl, the line must turn on the pearl and continue straight through both immediate neighbors, which is why the disclosed solution turns at the upper-left black pearl and then runs straight up the left edge to close the loop. If the path crossed itself near a pearl, the pearl constraints would still be evaluated against the unique-cell rule and the route would be reported as invalid even before the score was finalized.
These local constraints are why "touching every clue" is not the same as solving a Masyu. A path that snakes through every pearl without crossing or revisiting could still fail the white-pearl rule by passing through a white pearl and then turning nowhere nearby, and could fail the black-pearl rule by turning on a black pearl and then bending again on one of its arms. The browser's closure test checks both, which is why the clean twelve-step perimeter wins and any other accepted configuration still has to satisfy the same six closure conditions: unique body cells, full clue coverage, collinearity around the white pearl, a neighboring turn at the white pearl, a separate turn on the black pearl itself, and straight arms on both sides of the black pearl.
Common Mistakes With Loop and Pearl Constraints
Most failed attempts on the keyboard board fall into one of three families of mistake, and understanding each one clarifies why the simple-loop rule and the pearl rules cannot be negotiated.
The first family is closing too early. If the path returns to the start cell while one or more pearls are still unvisited, the browser reports a closed route but flags it as invalid because the clue-coverage check fails. The fix is to lengthen the route so that every marked pearl sits on the line before closure, which on this four by four board means the full twelve-step perimeter.
The second family is bending on the wrong pearl type. Turning on the white pearl in the top row or passing straight through the black pearl in the upper-left corner both fail at closure, because the white pearl would no longer have a "straight through" segment and the black pearl would no longer have a turn on its own cell. The fix is to memorize the disclosed geometry: straight on white, turn on black.
The third family is trying to shortcut the perimeter with a diagonal or a skip. Because the route moves only between orthogonally adjacent cells, a diagonal move is not part of the accepted input, and a jump over an empty cell is treated as leaving the grid, which the keyboard interface handles silently. Any attempt to skip a cell forces the path to revisit or branch later, so the disclosed perimeter is also the twelve-step solution that fits without violating the simple-loop rule.
Recovery Controls When the Path Gets Stuck
Because the loop cannot cross or reuse cells, getting stuck is a normal part of the early attempts on this board. The recovery controls are deliberately small so they can be exercised without leaving the page. Restart resets the cursor, empties the path, restores the 1,000-point starting score, and clears the solved status, which is the cleanest way to recover from a dead end without reloading.
Escape twice within 400 milliseconds reveals the shared simulated spreadsheet boss screen, and repeating the double press returns to the unchanged puzzle underneath. The boss screen does not pause a timer, because there is no timer on this board, and it does not save partial state, because the only persisted value is the best completed score held in this browser's localStorage. That score is never uploaded or attached to an account, so closing the tab keeps the best run in place and reopening the browser will surface it again on the next completion.
For readers who enjoy this kind of single-loop deduction, the related games in the same family use similar non-crossing constraints on different clue shapes. Shingoki Puzzle layers number and segment-sum clues onto a loop, while Pipes Puzzle rotates fixed tiles until they form one connected network without a closed loop, which makes the contrast with Masyu's mandatory single loop easier to feel.