Sudoku Solver

Extreme

X-Cycle (Discontinuous)

A single-digit chain that forms a loop with a contradiction, forcing an elimination.

X-Cycle (Discontinuous) is a powerful chain strategy that tracks a single digit through the grid. Unlike XY-Cycle which uses multiple digits, X-Cycle focuses on just one number.

The "discontinuous" part means the chain has a break that creates a contradiction—proving a candidate must be eliminated.

Interactive Example

1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
6
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
5
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
6
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
8
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9

Click "Apply Logic" to see the strategy in action.

The Core Idea

An X-Cycle follows one digit (like 5) through chains of: - Strong Links (=): If A is OFF, B must be ON (only 2 places for the digit in a unit) - Weak Links (-): If A is ON, B must be OFF (they see each other)

When a chain loops back to its starting cell with the opposite state, we have a contradiction.

Understanding Strong vs Weak Links

Link Type Symbol Meaning Example
Strong = Exactly 2 places for digit in unit If Row 1 has 5 in only C1 and C8
Weak - Cells see each other Same row, column, or box

Key Rule: Strong links alternate in BOTH directions. Weak links only work one way (ON→OFF).

Real Example Explanation

In the example above, we track digit 5:

The Chain: R9C4=5(ON) → Start: Assume R9C4 is 5 -R9C1(OFF) → Weak: R9C1 can't be 5 (same row) =R6C1(ON) → Strong: R6C1 must be 5 (only 2 places in column) -R6C8(OFF) → Weak: R6C8 can't be 5 (same row) =R1C8(ON) → Strong: R1C8 must be 5 -R1C6(OFF) → Weak: R1C6 can't be 5 (same row) =R2C4(ON) → Strong: R2C4 must be 5 -R9C4(OFF) → Weak: R9C4 can't be 5 (same column)

The Contradiction: - We started with "R9C4 = 5 (ON)" - We ended with "R9C4 = 5 (OFF)" - This is impossible! The assumption was wrong.

The Elimination: Remove 5 from R9C4.

Types of Discontinuous Loops

There are two types based on where the "break" occurs:

Type 1: Two Weak Links Meet

  • The chain has two consecutive weak links at the break point
  • A candidate seeing BOTH ends of the break can be eliminated

Type 2: Two Strong Links Meet

  • The chain has two consecutive strong links at the break point
  • The break point cell must BE the digit (it's forced ON)

How to Spot It

  1. Pick a Digit: Choose a number (1-9) to track.
  2. Find Strong Links: Look for units (row/col/box) with exactly 2 places for that digit.
  3. Build the Chain: Alternate strong and weak links.
  4. Check for Loop: Does the chain return to the starting cell?
  5. Check the Break: Are there two consecutive weak links (Type 1) or strong links (Type 2)?
  6. Make Elimination: Apply the appropriate elimination rule.

Comparison with Related Strategies

Strategy What It Tracks Link Types
X-Cycle Single digit Strong + Weak
Simple Coloring Single digit Strong only
XY-Cycle Multiple digits Bivalue cells
XY-Chain Multiple digits No loop

Related Strategies