Can the path finish there?
Pick two squares. Can you travel from one to the other and visit every square exactly once? Check what the alternating colors rule out, then see what an actual route proves.
Watch two endpoint attempts
See why A1 to B1 fails, then follow a complete route from A1 to C3.
Read the walkthrough
Nine squares contain five cream and four blue. Each move changes color, so a path starting on cream must end on cream after eight moves. B1 is blue. C3 is cream. One complete route runs A1, B1, C1, C2, B2, A2, A3, B3, C3.
Why shade the squares?
A horizontal or vertical step always changes color. On nine squares, the start and finish must both use the color that appears five times. On a board with equal numbers of both colors, the ends must have opposite colors.
Passing that check is only a first test. Try the ladder with B1 and B2 as the ends. Both colors are represented, yet those two middle squares separate the left side from the right. A single path between the middle ends cannot cover both sides without revisiting an end.
How many full-grid paths are there?
On the 3-by-3 board, only 10 of the 36 possible endpoint pairs work. Each of those ten pairs has exactly two paths. Try A1 to C3 in the checker, then see if you can draw the other route.
| Board | Endpoint pairs | Pairs that work | Full paths |
|---|---|---|---|
| 3 × 3 | 36 | 10 | 20 |
| 4 × 3 | 66 | 29 | 62 |
| 3 × 2 | 15 | 8 | 8 |
How the paths were counted
Tiles have fixed addresses. A path moves to a neighboring tile horizontally or vertically and visits every tile once. An endpoint pair counts once regardless of order. Walking the same path backwards also counts once. Reflections or rotations that use a different set of edges count as different routes.
For every pair we enumerated legal moves, then independently verified the count by working backward through subsets of visited tiles. These counts apply to the empty rectangular boards shown here. Ordered sum targets in the separate iPhone game introduce another constraint.
Download counts for all 117 endpoint pairs (CSV) · Download the reproduction script (Python)
Add ordered sums in Sumthread
This original lab isolates the path rule. In Sumthread, a single orthogonal path also has to reach its target sums in order. Try the daily browser puzzle or explore the iPhone game.
View Sumthread on the App Store (Canada)