| C10 | C21 | C30 | C42 | C51 | |
|---|---|---|---|---|---|
| R11 | |||||
| R20 | |||||
| R31 | |||||
| R40 | |||||
| R52 |
The displayed tents match every count and do not touch. Is this a valid finish?
Make a prediction before opening the reasoning.
Compare the predictions
- Yes: the counts and spacing all work.
Not supported. The two top trees both need the same tent at R1C4.
- No: two trees compete for one tent.
Supported. R1C3 and R2C4 have only R1C4 available between them.
- No: two tents touch diagonally.
Not supported. The tents satisfy spacing. The failure is their inability to pair distinctly with all trees.
Predictions stay on this page; they are not saved or sent.
The position and the question
The row clues are 1, 0, 1, 0, 2 and the column clues are 0, 1, 0, 2, 1. Four tents match those totals. No two tents touch, even diagonally. Each tent is beside a tree, and every tree has at least one neighbouring tent.
One rule still needs a stronger check: the trees and tents must admit a one-to-one pairing through side adjacency. A tent can touch two trees geometrically, but it cannot serve as the assigned tent for both. We need a set of distinct partners, not merely a neighbour for each item.
Show the reasoning
Inspect the two top trees together
Tree R1C3 has only the tent at R1C4 as a possible partner. Tree R2C4 also has only the tent at R1C4: R3C4 is grass and its other neighbours hold no tents. Both trees individually pass an adjacency check, but together they need two distinct tents and have only one available.
Explain the failure before moving anything
The lower tents cannot rescue the top pair. A matching uses only side-adjacent tree–tent pairs; it cannot transfer a distant tent across the board. This is a complete impossibility proof for the displayed arrangement, despite all row and column totals being correct.
Repair two placements while preserving the clues
Move the tent at R3C5 to R3C4 and the tent at R5C4 to R5C5. Each remains in the same row. Column 4 gains one tent and loses one; column 5 loses one and gains one. Therefore every row and column clue stays satisfied. The new tents also remain mutually non-touching.
Required tent counts; T marks a tree C10 C21 C30 C42 C51 R11 R20 R31 R40 R52 Repair two placements while preserving the clues Write the distinct pairing
Pair R1C3 with R1C4, R2C4 with R3C4, R5C3 with R5C2 and R4C5 with R5C5. Every pair shares a side and every tent appears once. A complete check of the three non-touching count-valid layouts for these trees and clues finds this is the only layout with a full pairing.
The tempting shortcut
If every tree and every tent has some neighbour, the pairing rule is satisfied.
Where the shortcut breaks
Two trees can both point to the same sole tent. Individual adjacency checks do not ensure distinct partners.
A better decision
For a small suspicious group of trees, count the different tents the group can use; fewer tents than trees proves a pairing failure.
Changed conditions
Could you repair the arrangement by moving only R3C5 to R3C4?
Check the changed-condition answer
That supplies the top pair with distinct tents, but breaks the column counts: column 4 rises from 2 to 3 and column 5 falls from 1 to 0. The second move from R5C4 to R5C5 is needed to preserve both column clues.
Take this back to your next game
When objects require distinct partners, checking each object separately can miss a group-level shortage.
How the claim was checked
Enumerate four-tent subsets of non-tree squares; enforce counts, non-touching, both-direction adjacency and exhaustive bipartite matching.
- Displayed arrangement passes all local checks but has no full matching
- Top two trees have only one possible tent partner
- 3 non-touching count-valid layouts; 1 full matching layout
Read the fixed positions and supporting proof data. Computational checks establish only the claim and position stated here. They do not establish a player difficulty rating or make an unproved strategy optimal.
Playfield Arcade · Updated 4 October 2026 · Original case design, AI-assisted writing and code, with reproducible position checks. Our method · Report a correction