Why this turn?
Four deductions in Turning a corner. Green squares are newly placed numbers.
Only one way out.
Find 1 in the bottom row. The square above it already holds 6; the square to its left holds 16. Neither can be 2. There is no square below the grid.
That leaves the square on the right for 2. This is a deduction from the printed clues, not a guess about the whole path.
Three reaches four.
From 2, going up would meet the printed 5 too soon. Going left revisits 1. The right-hand square must be 3.
Now 4 sits immediately above 3. Follow the printed 4, 5 and 6 in order. A clue is part of the trail, not a square to skip.
Fourteen has two neighbours.
The printed 13 and 15 are two steps apart vertically. To connect them in exactly two moves, 14 belongs in the square between them.
A side detour needs extra moves. You can write this deduction on paper before reaching 13 from the start.
Leave one square for 24.
Along the top row, 23 and 25 have one square between them. The difference is two, so there are exactly two moves available. Put 24 between the clues.
On paper, mark these certain numbers firmly and trial numbers lightly. Keep the answer sheet apart from the unsolved puzzle.
A legal next step can still lead nowhere.
Two more readings of the same original puzzle. Rows and columns below are counted from the top left, starting at 1.
Seven must go above six.
The printed 6 is at row 4, column 3. Its left neighbour is the printed 15, its right neighbour is the already-used 5, and below is the starting 1. Only row 3, column 3 remains for 7.
Keep that certain 7 when exploring later turns. Undoing a failed trial need not erase the deductions that came before it.
Twelve cannot reach thirteen from here.
Try this sequence only as an illustration of a failed branch: 8 at row 2 column 3; 9 at row 2 column 4; 10 at row 2 column 5; 11 at row 3 column 5; 12 at row 3 column 4.
Each of those moves touches the previous square on an edge. But the printed 13 is at row 2, column 2. From the trial 12, it is one row and two columns away: at least three moves. There is only one number step left. This branch cannot continue.
Erase the trial 8–12. Keep 1–7 and the separate certain 14 and 24 deductions from above. The failure rejects this complete trial; by itself it does not prove that every possible continuation after the same first turn fails.
A small distance check before a long detour
For an orthogonal grid, count the row difference plus the column difference between your current square and the next fixed clue. If that minimum exceeds the number difference, the clue is unreachable in time. In the wrong trial, 1 + 2 = 3, while 13 − 12 = 1.
Passing the distance check is only a necessary condition. Occupied squares, other fixed clues and the requirement to visit every square can still block the route. A nearby clue does not certify the rest of a puzzle.
On paper, put a small pencil mark beside the first uncertain number. In the browser, use Undo back to that point. Before you ask for a clue, describe the obstacle precisely: “I cannot reach 13 in the remaining move” gives you something to test; “the puzzle is broken” does not.
You do not have to solve a trail in one sitting. If you make a trial move, leave an arrow at its first square. When it fails, erase back to that arrow instead of losing every deduction. These four diagrams reveal parts of this puzzle, not its whole answer.
Try the remaining turns →