Loop Trace strategy: check the dot before completing the cell.
Use a zero cell and a three to start a new board, compare a tempting cell-completing branch, then alternate dot checks with cell counts to justify a twelve-line solution.
A line can complete a cell number and still be wrong. The cell counts its bordering sides; a dot checks how those sides meet. A finished dot must have zero or two lines, never one, three or four. You need both kinds of check, followed by the single-loop check for the whole drawing.
This lesson uses a new three-by-three board rather than the first lesson’s square loop. We will start with a zero beside a three, test a misleading choice, and justify each continuation. The worked route uses twelve line changes. It is a reproducible route, not a minimum-edit contest or a difficulty rating.
Load the separate strategy board
Choose Practice this example beside the article, or Load strategy board in the game. Confirm Edits 0 and these clues, read left to right and top to bottom:
2 1 2 2 0 2 2 3 2
For a cell coordinate, R3C2 means the bottom-middle cell. For a dot coordinate, row and column count the four dot rows and four dot columns. H3-1 joins dots in dot row 3, columns 1 and 2. V2-1 joins dot rows 2 and 3 in dot column 1. The H and V identifiers describe edges, not cells.
Keep unused-edge deductions on paper for this route. You may enter crosses in the game, but each mark change adds an edit; our checkpoint counts assume you draw only the listed lines. An unmarked edge can still be excluded by your reasoning. The game does not store that reasoning or choose a correct next edge.
Let the zero give the three its missing information
The centre zero, R2C2, excludes H2-2, H3-2, V2-2 and V2-3. Its four sides must remain unused. Now inspect the bottom-middle three, R3C2. Its top side is H3-2, already excluded. Therefore its other three sides must all be lines: H4-2, V3-2 and V3-3.
Draw those three edges in that order. At Edits 3, they make one U-shaped path. Exactly two cell clues match: the centre zero and the bottom-middle three. There are two open ends and no branching dot. The path is not yet a loop.
A tempting edge passes one check and fails another
The bottom-left cell, R3C1, currently has one line, V3-2, and needs two. It is tempting to add its bottom side H4-1. Before doing so, inspect the dot at the right end of that edge: dot row 4, column 2. H4-2 and V3-2 already meet there. A third line would create a branch.
Try the mistake as a separate experiment. From the three-edge checkpoint, add H4-1, then choose Check loop. Predict Edits, matching cell clues, open ends and branching dots before revealing the answer. This interface permits a wrong line while you work; acceptance is not a hint that the line belongs in the solution.
Reveal the four-edit branch and recovery
Edits becomes 4. The bottom-left two now matches, so three of nine cell clues match. There are four lines in one group, three open ends and one branching dot. The new left endpoint adds an open end, while the existing bottom dot changes from degree two to degree three. The cell total improved, but the drawing cannot be the required nonbranching loop. Undo once restores Edits 3, the U-shaped path, two matching clues and two open ends. H4-1 is again unmarked; remember that the dot rule excludes it.
Alternate between dots and cell totals
Return to Edits 3 before following this route. The centre zero exclusions still hold in your notes. At a dot with one line and only one available continuation, draw that continuation. At a cell whose target is reached, exclude its other sides. At a used dot with two lines, exclude every other incident edge.
- The zero excludes the top of R3C2, so its three remaining sides are H4-2, V3-2 and V3-3. These are the three lines already drawn at the checkpoint.
- At dot row 3, column 2, V3-2 is a line, while H3-2 and V2-2 are excluded by the zero. Draw H3-1 to give that dot its second line. R3C1 now has its required two sides, H3-1 and V3-2, so exclude H4-1 and V3-1. At dot row 3, column 1, only V2-1 can continue H3-1. Draw it.
- R2C1 now has two lines, H3-1 and V2-1; its right side V2-2 is excluded. Its top side H2-1 must also stay unused. At dot row 2, column 1, V1-1 is the only continuation of V2-1. Draw V1-1, then H1-1 at the top-left corner dot.
- R1C1 has its two sides, V1-1 and H1-1, so exclude V1-2. Dot row 1, column 2 then forces H1-2. That line satisfies the top-middle one, R1C2; with H2-2 and V1-2 excluded, V1-3 must also be unused. Dot row 1, column 3 therefore forces H1-3.
- The top-right corner dot forces V1-4. R1C3 now has its required two sides, so exclude H2-3. Dot row 2, column 4 forces V2-4. R2C3 needs two lines, but its top H2-3 and left V2-3 are excluded; V2-4 and H3-3 must be its two lines. Draw H3-3 to join the remaining ends.
The last line closes the single path. The bottom-right two has H3-3 and V3-3, so H4-3 and V3-4 remain unused. All the exclusions and line decisions agree; no extra small loop or branch is needed.
Use checkpoints to find a mistaken deduction
| Edits | Matching clues | Line groups | Open ends |
|---|---|---|---|
| 3 | 2 of 9 | 1 | 2 |
| 5 | 4 of 9 | 1 | 2 |
| 9 | 6 of 9 | 1 | 2 |
| 12 | 9 of 9 | 1 | 0 |
A checkpoint is a comparison after a justified step, not a reason to add an arbitrary line until a counter matches. If your result differs, inspect the edge labels in the ledger or Undo the last change. Retry this board starts this same strategy example again. Loading a different lesson replaces the clues, while refreshing its fixed practice address clears the marks and restores this board.
Reveal the complete edge ledger and final self-check
Each row below lists line counts. A zero means unused; it does not require a cross mark in the game. Horizontal rows have three positions, vertical rows have four.
H1: 1 1 1 H2: 0 0 0 H3: 1 0 1 H4: 0 1 0 V1: 1 0 0 1 V2: 1 0 0 1 V3: 0 1 1 0
The bordering counts reproduce 2 1 2 / 2 0 2 / 2 3 2. The twelve lines form one connected group with no open ends or branching dots. Undo after the win removes H3-3: Edits returns to 11, seven cell clues still match, and two open ends remain. The right-middle and right-bottom cells each lose a bordering side. Replacing H3-3 completes the loop again.
What the method does and does not prove
These local checks solve this prepared board without choosing an unproved branch. That does not mean every numbered-cell loop puzzle can be finished by these checks alone, or that checking dots first is always the best order. When neither a cell nor a dot forces a decision, leave the alternatives open instead of treating a preference as a proof.
Always check the whole drawing too. Matching all numbers does not, by itself, replace the requirement for one closed, nonbranching loop. Our new clue set was independently checked for a unique valid loop. That is a property of this example, not a promise about difficulty or general reasoning ability.