Logical Solvers

U-Bahn rules. Draw one totally connected network through the centers of some unshaded cells. The network may branch or turn but may never dead-end: every used cell connects in 2, 3, or 4 directions. A clue outside the grid gives the exact count of a line shape in that row or column, regardless of the shape's rotation: ┼ cross (4 connections), ├ branch (3), ─ straight (2 opposite), └ turn (2 adjacent) — or, in the outermost band, of · empty cells the network never visits (shaded cells count as empty).

Entering clues. Clues are arranged the traditional U-Bahn way: the five rows above the grid hold column clues and the five columns on the left hold row clues — one band per shape, in the order shown by the icons on the corner diagonal (each icon labels the band to its right and the band below it). Leave a box empty for "no clue"; enter 0 to require zero of that shape. Click any grid cell to shade it (shaded cells can’t be used by the network).

Solve finds one valid network and tells you whether it is unique. True candidates proves, for every individual segment, whether it appears in every solution (solid blue), in some but not all (dashed gray), or in none (a small × on that border) — each segment is tested independently, so the result is exact even when there are far too many solutions to enumerate. The solid network is the guaranteed partial solution: it can legitimately contain loose ends and partial pieces, because a cell’s remaining connections may differ from solution to solution. Tinted cells are used in every solution (even when no single segment through them is fixed); × marks cells used in none. Amber dashed segments could not be proven either way before the time limit — everything else shown is proven. Raise the limit and re-run to resolve them.

Take step makes exactly one logical deduction at a time, sudoku-solver style: it keeps pencil marks for each cell (remaining shapes among L turn, I straight, T branch, X cross, · empty), draws borders it can prove (line or ×), and explains each move. The full ladder of ideas it tries — from clue counting through cross pigeonholes, connectivity (sealed loops, unreachable regions, forced bridges), segment capacity, and the parity tricks of the classic solving guides — is listed under Step deduction types beside the board, with the idea behind the last step highlighted. When no human-style rule applies it falls back to an exhaustive test of a single border and says so. Steps reset automatically whenever you change clues or shading.

Digits 1..D, at most once per row and column; shaded cells hold no digit. One clue box per group, in reading order. A single 0 = the line is entirely blank; all boxes empty = unclued.

? = one unknown digit (so ? is 1–9, 1? is 10–19, ?? two digits); # = a group of any sum. Letters are crypto digits 0–9 — each letter one digit, all letters different, no leading zeros.

The Custom values field replaces digits 1…D with a palette, e.g. 1,2,3,4,5,5,6,7,9,9: no 8s, two 5s and two 9s per line. Negatives and double-digit values work; clue sums may then be negative, written like -7.

Alien = the clues are written in an unknown number base (2–31) that the solver must determine — every clue digit and cipher letter is a digit of that base, so letters can stand for values of 10 or more. A base digit needing two decimal characters is written with dots: 11.3 is the two-digit numeral with digits 11 and 3. The base box beside the cipher tracks the remaining candidates.

A38. Draw a directed orthogonal loop through every non-number cell, beginning and ending at the black circle and passing through every gray pass cell. The loop cannot branch, cross, or revisit a cell.

Each number N identifies the Nth of its up to eight neighbouring route cells reached by the traveller. Identified cells grant passes; if several clues identify the same cell, it still grants only one pass. A gray cell is counted in neighbour order but can never grant a pass. The traveller starts without a pass, can carry one, and spends it on entering a gray cell.

Choose Start, Pass, or Number and paint the grid. Clicking a cell with its active tool again restores an ordinary Route cell. In a number cell, type digits 1–8 separated by spaces — or ? marks for extra granting positions whose numbers are unknown — each ? is one more granting position, all distinct from each other and from the listed numbers (the solver resolves them to the lowest numbers that work). Backspace removes entries normally, and one more Backspace in an empty number cell removes the cell itself (clicking an empty number cell with the Number tool active does too).

Take step applies exactly one immediate human deduction to persistent pencil marks. It never calls or caches the exhaustive solver. A solid segment has been proved part of the route; an arrow also proves its direction; × excludes an edge. Solve and True candidates are separate exhaustive operations.

Cave. Shade cells so that all unshaded cells form one orthogonally connected cave. Every connected group of shaded cells must reach an edge of the grid: shaded cells are the outside of the cave, and no shaded pocket may be enclosed.

A number lies in an unshaded cell and gives the total number of consecutive unshaded cells visible horizontally and vertically, including its own cell. A shaded cell stops sight in that direction. A ? is an unshaded clue cell whose number is not given. A valid Cave has at least one shaded cell.

Optional restrictions. No 2×2 shaded and No 2×2 unshaded forbid monochrome 2×2 blocks of the selected colour.

Twilight. Number cells may be shaded or unshaded. An unshaded number is the usual Cave sight clue. A shaded number instead gives the exact size of the orthogonally connected shaded region containing it. In Twilight, values from 1 through the number of grid cells are accepted; a value outside the ordinary sight range necessarily takes the shaded interpretation. A ? may take either colour and supplies no count.

Click a cell and type a clue, or ?. Backspace clears it. Solve shows one solution and checks uniqueness. True candidates marks cells that are shaded or unshaded in every solution. Take step makes one named deduction, while Previous step and Full solve path navigate the same persistent logical path used by the other tabs.