Queens Solver

LinkedIn Queens — information-gain analysis

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preview
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Grid size
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Deduction steps
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Cells resolved
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Entropy (bits)
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Method
Solved board
Step-by-step log
Cell-info Pareto — contribution by rule
Entropy Pareto — reduction by rule
Cumulative information gain

How it all works

The puzzle

LinkedIn Queens is a constraint puzzle played on an n × n grid. The grid is divided into n coloured regions — contiguous groups of cells that each share one colour. Your goal is to place exactly one queen (♛) in every region so that no two queens share the same row, the same column, or touch each other — even diagonally. On a 5 × 5 board you place 5 queens; on an 11 × 11 board, 11.

How the solver thinks

Rather than guessing and checking, the solver works like a logician: it applies a prioritised set of deduction rules, each one narrowing down the possibilities until every queen is placed — usually without ever needing to guess. Each rule fires only when it can prove something new. Here is what each one does:

Region singleton

A region has been narrowed down to exactly one cell where a queen can legally go. The solver places the queen there — no choice involved. Row and column singletons work the same way but trigger when an entire row or column has only one candidate cell left.

Elimination

If placing a queen in a particular cell would make it impossible to place a queen in some other region at all — every candidate in that region would be blocked — then that cell is ruled out. This is the solver's sharpest deduction: it looks one move ahead and eliminates anything that would create a dead end, without actually trying it.

Region forced row / col

All remaining candidates in a region happen to fall on the same row (or column). That row (or column) is now "claimed" by this region, so every other region's candidates on that row (or column) can be eliminated.

Double-Block

If placing a queen in a specific cell would force two different regions to both use the same row or column — leaving them with no way to coexist — then that cell is eliminated. This catches conflicts that wouldn't directly kill any single region but would create an impossible pairing between two regions downstream.

Triple-Check

When three adjacent rows (or columns) contain all the candidates for exactly three regions and no other regions have candidates in those lines, then those three rows (or columns) are "reserved" for those three regions. Any other candidates in those lines from different regions can be eliminated.

Lookahead

For small regions (2–5 candidates), the solver trial-places a queen in each candidate cell and fast-forwards the deductions. Any candidate that leads to a contradiction is quietly removed. Only candidates that survive all trials are kept. This is stronger than elimination but still fully logical — no guessing.

Search

A last resort, rarely needed. The solver picks the most constrained region, guesses a candidate, and recurses. If it hits a dead end it backtracks and tries the next candidate. A puzzle that requires search is genuinely harder than one solved by pure deduction.

Reading the analysis

Grid size
The board dimensions: n × n. Also tells you how many queens need to be placed and how many colour regions exist.
Deduction steps
The number of times a rule fired and changed the board state. Fewer steps generally means the puzzle had a tighter logical structure — each move unlocked more consequences.
Cells resolved
Total cells whose fate was determined (queen placed or candidate eliminated). For an n × n board the maximum is n².
Method
Pure deduction means every queen was placed through logical rules alone. Search means the solver had to guess at least once — the puzzle had ambiguous states that logic alone couldn't resolve.

Information gain & entropy

The numbers and charts on this page don't just show what the solver did — they measure how much each step mattered. That measurement comes from information theory, and it's worth understanding why.

Why measure information at all?

Two solvers can both find the correct answer, but one might do it in a way that's deeply insightful and another in a way that's mostly grinding. Raw step count doesn't capture this. A single elimination move that wipes out 32 candidates across the board is far more powerful than 10 singleton moves that each place one obvious queen. Information gain lets us quantify that difference.

Entropy (bits)
Entropy is a measure of how much uncertainty remains in the puzzle. Think of it as the total number of yes/no questions you'd need to ask to pin down where every queen goes. At the start, when almost anything is possible, entropy is high. When the puzzle is solved and every queen is placed, entropy is zero. The unit — bits — is standard in information theory: one bit is the information you gain from one fair coin flip. A puzzle with 30 bits of starting entropy is, in a precise sense, roughly 2³⁰ (about a billion) times harder than a puzzle with zero bits remaining.
Δentropy (entropy change)
How many bits of uncertainty a single step removed. A large negative Δentropy means that step dramatically narrowed the solution space — it was a high-leverage move. In the step log, look for the biggest drops: those are the moments the puzzle "cracked open."
+Cells / Total (cell certainty)
A second, more intuitive metric running alongside entropy. +Cells is how many cells were newly determined by that step (queen placed or candidate eliminated). Total is the running count. When Total reaches n², the puzzle is fully resolved. Cell certainty and entropy usually move together, but not always — sometimes a single deduction resolves few cells but slashes entropy because it removes a high-uncertainty region from the board.
Pareto charts
Named after the economist Vilfredo Pareto, who observed that roughly 80 % of effects come from 20 % of causes. The two Pareto charts here rank each deduction rule by its total contribution — one by cells resolved, one by entropy reduced — and show the cumulative percentage. If one rule accounts for 60 % of the information gain, that's the engine of the solve. The cumulative percentage lets you see how quickly a small number of rules does most of the work.
Cumulative information gain
The rising curve at the bottom plots total cells resolved after each step. A steep early climb means the first few moves were high-leverage — each one unlocked a cascade of consequences. A curve that flattens and then climbs again in a staircase pattern means the solver hit stretches where progress was slow, then found a breakthrough. The shape of this curve is a fingerprint of the puzzle's difficulty and structure.

The short version

Entropy tells you how hard the puzzle was. Δentropy tells you which moves mattered most. The Pareto charts tell you which types of reasoning did the heavy lifting. Together they turn a solved puzzle into a story: where was the real challenge, and how did the solver meet it?