Open Rules

A complete one-cell arrow adds effective shaft values to the bulb value. A partial arrow adds face digits to at most the bulb’s face-digit number. A bulb with several cells forms a decimal number in the listed order.

The base and tip digits give the row and column of a cell, counted from 1. That cell contains their sum. Every arrow names a different coordinate pair.

Digits between the two ends add up to the sum of the end digits.

Count along the ray by the marked digit to find its target. The mode states whether the target is greater, smaller, equal to the source, or the first 5.

Cells a chess king move apart contain different digits.

Cells a chess knight move apart contain different digits.

Divide the board into connected regions of the stated size. Each region contains different digits. Borders separate regions; joins put cells in the same region.

Each sufficiently long diagonal either stays in one digit class or repeats complete windows of all classes.

Digits on the stated main or opposite diagonal are different.

Every listed group has the same total.

Every occurrence of the marked digit has the stated above digit immediately above it, or the stated below digit immediately below it.

At least one listed square contains 1 to 9 once each. Its rows, columns and both diagonals each sum to 15.

Electron digits around each atom add up to its stated total. All cells at or below the electron threshold are electrons; higher digits may also be chosen. Touching atoms share the same electron choices.

Unmarked neighbouring pairs fail every listed relationship. The scope, direction and exceptions state exactly which pairs are covered.

Each digit and the digit opposite it after a half turn add up to one of the listed totals.

For the listed digits and chess move, prohibit either equal digits or consecutive digits in the related cells.

Every row, column and standard 3×3 box contains the digits 1–9 exactly once.

Choose modified cells according to the stated group counts. Their values are their digits times the factor, plus the offset. The same choices apply to every rule that reads values.

Every two-by-two window contains every stated digit class.

Shade the board so each colour forms one connected group and every 2×2 contains both colours.

Each cage total equals its expression. Letters stand for whole numbers and keep the same value throughout the rule. The distinct parameter gives different letters different values.

The totals of the two cages obey the stated relationship.

Construct the hidden cages around their clues. Each cage has the stated size, includes its known cells, avoids free cells, and contains different digits adding to its total. Cages are connected and do not overlap.

Cage digits add up to the stated total value. Digits are different unless repeats are allowed. A cage without a total supplies the repeat rule alone.

Cage digits are different. The number of low odd digits equals the number of high even digits; low even and high odd counts also match.

Cage digits add up to the stated total value. Digits are different unless repeats are allowed. A cage without a total supplies the repeat rule alone.

Each occupied row of the cage contains only upper digits or only other digits. Every upper row is above every other row.

Each listed digit occurs exactly the stated number of times in the cage.

Multiplying the cage digits gives the stated product. The distinct parameter forbids repeats.

The cage total has the stated remainder after division by the modulus. The distinct parameter forbids repeated digits.

Cage digits are different. Adding any two cage digits, including a digit to itself, and taking the stated remainder never produces another cage digit.

No digit inside the vault appears in its listed neighbours. A stated sum fixes its total.

None of the listed digits appears in the marked cells.

All listed cells contain the same digit.

Each digit appearing in the circles tells you how many circles contain that digit.

Each marked digit is smaller than all its unmarked neighbours for a minimum, or larger for a maximum. Exhaustive marking also constrains every unmarked cell.

Choose the span by its stated extent. Every three consecutive digits in that span have different remainders when divided by 3.

A digit points to its own position in the stated row or column axis. The digit in the pointed cell gives the position of the original cell.

A white relation joins values one apart. A black relation joins values where one is twice the other.

The two values obey the stated relationship: a difference, ratio, total, multiple, inequality or shift.

Values in the marked cells have the stated parity: odd or even.

The marked digit counts its listed neighbours that are greater than it for a low gauge, or smaller for a high gauge.

The surrounding cells contain every listed digit, including the stated number of copies.

All marked digits belong to the same stated class: parity, remainder or alphabet third.

The marked digit is greater than every listed neighbour.

All digits in the listed cells are different. Declare each row, column and region that needs this rule.

Every marked cell contains one of the listed digits.

Every digit between the ends is strictly greater than the smaller end and strictly less than the larger end.

The different digits on the line form a consecutive run. Digits may repeat where the other rules allow it.

Neighbouring digits along the line differ by at most the stated maximum.

Every three consecutive digits contain one from each third of the ordered alphabet. On a two-cell line, the digits come from different thirds.

The two paths contain the same digits in corresponding positions.

Count positions from the marked end, starting at 1. A digit X in position N means the digit in position X is N.

The marked cells contain at most the stated number of different digits.

The line satisfies between the stated minimum and maximum number of the listed line rules.

The end digits differ by at least the stated gap. Every digit between them is below both ends or above both ends.

Every listed line has the same average digit.

Every three consecutive digits have different remainders when divided by the stated modulus. On a two-cell line, the remainders differ.

No two digits anywhere on the line are consecutive. The distinct parameter also forbids repeats.

The values along the line read the same forwards and backwards.

Digits alternate between odd and even along the line.

Digits strictly increase from bulb to tip. Count outward along the ray by the tip digit: that cell contains the bulb digit.

Prime and composite digits alternate along the line. On a standard board, primes are 2, 3, 5, 7 and composites are 4, 6, 8, 9.

Each listed segment of the line has the same total value. Crossing a region boundary begins a new segment.

The line contains different values that form a consecutive run, in any order.

Digits on the line cannot be 5. Within a region, neighbours are both odd or both even and both below or both above 5. At each region crossing, exactly one of those relationships changes.

Digits never decrease from the bulb to the tip. Equal neighbours are allowed.

Neighbouring digits differ. Digits rise or fall along the line, changing direction only at 1 or 9.

Values strictly increase from the bulb to the tip of the thermometer.

Neighbouring values along the line differ by at least the stated gap.

Neighbouring values along the line differ by at least the stated gap.

Pairs equally far from the centre have the same sum. On an odd-length line, that sum is the centre digit.

For each listed total, two digits sum to that total and all digits strictly between them also sum to it.

Digits along the listed ray add up to the clue. Repeats are allowed by this rule.

Count along the ray by the clue digit. That target digit plus the clue equals the cage total. Cage digits are different.

Count from the clue by the first digit. The cell reached contains the stated digit.

The listed digits appear as a consecutive block of cells, in exactly that order.

The listed cells contain every stated digit, including its stated number of copies.

Count each digit once when a neighbour along the line differs by 1. These digits add up to the clue.

The clue gives the sum of digits strictly between the two crusts, counted from the clue side. State the crust digits and their occurrence policy.

Read from the clue. A digit is visible when it is greater than every digit before it. The clue counts visible digits.

The first digit gives how many cells to count from the clue. Those digits add up to its total. A stated count replaces the first digit.