The Information-Gain Rule: How to Choose Puzzle Moves That Reveal the Most

The Best Move Is the One That Teaches You the Most

The information-gain rule is simple: when several puzzle moves look possible, prefer the move that is likely to reduce the most uncertainty. Instead of asking only, “Could this be correct?” ask, “Whatever happens next, how much will I learn?” This approach turns guessing into investigation and helps you make steady, purposeful progress.

A tempting move may advance one small part of a puzzle. A high-information move can clarify several parts at once, eliminate many candidates, or expose a hidden pattern. It may not solve anything immediately, but it makes the next decision easier—and often starts a chain of deductions.

What “Information Gain” Means

Information gain is the reduction in uncertainty produced by a question, test, clue, or action. The idea comes from information theory and is also used in decision-tree algorithms, where a useful test divides uncertain data into clearer groups. IBM’s introduction to decision trees and information gain explains this as the difference between uncertainty before and after a split.

Imagine that a secret number is somewhere from 1 to 100. You can ask one yes-or-no question.

• “Is it greater than 90?” creates groups of 90 and 10 possibilities.
• “Is it greater than 50?” creates two groups of 50.

The second question is normally more informative because either answer removes about half the possibilities. The first question could be helpful, but its outcomes are uneven: one answer still leaves 90 candidates.

Puzzle solving follows the same principle. A strong move tends to divide the remaining possibilities into reasonably balanced groups or remove a large group outright.

Think Like a Detective, Not a Guesser

A random guess tests one possibility. An investigative move tests an entire family of possibilities.

Suppose a word puzzle has 12 possible answers remaining. Guessing a word that differs from the others by only one letter may confirm or reject that single distinction. Testing a word containing several uncertain letters could reveal much more through one round of feedback.

The same contrast appears across many puzzle types:

• In Sudoku, filling a safely determined cell at the intersection of a crowded row, column, and box may eliminate candidates in all three areas.
• In a crossword, answering a clue with several crossings can provide letters for multiple unfinished words.
• In a logic grid, processing a clue that connects three categories may create more eliminations than a clue affecting one pair.
• In Minesweeper, analyzing a numbered square shared by several unknown cells can clarify an entire border.
• In a jigsaw, locating a piece with a distinctive shape and image feature can establish an anchor for a whole region.

These examples share one feature: the move has reach. Its consequences spread beyond the place where it is made.

The Three Qualities of a High-Information Move

You do not need to calculate mathematical entropy while enjoying a puzzle. Instead, evaluate moves using three practical qualities.

1. It Affects Many Possibilities

Count how many candidates, cells, clues, pieces, or paths the move could influence. A crossword answer with five crossings generally has more reach than an isolated answer with one crossing. A Sudoku cell connected to many unresolved candidates may produce more follow-up deductions than one in an almost completed area.

This is easier when your remaining options are visible. The Puzzles Arcade guide to constraint mapping explains how to organize what is possible, impossible, and confirmed before choosing where to focus.

2. Its Possible Outcomes Are Useful

A move is informative when every likely result tells you something meaningful.

Consider a game in which eight objects could contain a hidden key. A test that separates them into groups of four and four is powerful. Whether the test succeeds or fails, four possibilities disappear. A test separating them into groups of seven and one is less reliable: one result is excellent, but the other barely narrows the search.

This is sometimes called a “split-half” approach. You do not need a perfect half-and-half division, but balanced outcomes usually provide more dependable progress.

3. It Is Safe Enough to Test

Information has value only if obtaining it does not ruin the puzzle. Before experimenting, consider whether the move can be undone.

Pencil marks, temporary tile arrangements, written assumptions, and games with reliable undo functions make testing safer. When moves are permanent—or consume limited guesses, lives, hints, or time—the required information gain should be higher.

Use the reversibility rule to separate low-risk experiments from commitments that could leave you stuck.

A Simple Scoring Method for Your Next Move

When you are stuck between several options, give each one a quick mental score using four questions:

1. Reach: How many unresolved elements could this affect?
2. Split: Will the possible results divide the remaining options clearly?
3. Safety: Can I reverse the move if the idea fails?
4. Cost: How much time, space, or limited opportunity does it consume?

You can remember this as:

High reach + clear split + good safety − high cost

This is not a formal equation. It is a practical comparison tool.

For example, imagine you can attempt either of two crossword clues. Clue A is a difficult ten-letter answer with six crossings. Clue B is an easier four-letter answer with no confirmed crossings and provides only one new crossing letter.

Clue B may be faster to answer, so it could still be the right choice. However, Clue A has greater potential information gain. Even a partial answer, likely ending, or confirmed letter pattern might unlock several surrounding clues.

The rule does not say to choose the hardest move. It says to choose the move with the best information return for its cost.

How the Rule Works in Popular Puzzles

Word and Code-Guessing Games

In feedback-based word games, early guesses should usually test useful letters and positions rather than repeat information you already possess. Later, when only a few candidates remain, a valid “probe” word may distinguish between them more efficiently than guessing one candidate at a time.

Mastermind offers a famous mathematical example. Donald Knuth demonstrated a strategy for the standard four-position, six-color game that can determine the code within five guesses by selecting guesses that control the worst-case number of remaining possibilities. The MathWorld overview of Mastermind strategy describes this possibility-reducing approach.

The important lesson is not the exact algorithm. It is that a guess can be valuable even when it is unlikely to be the final answer, provided its feedback sharply separates the candidates.

Sudoku and Logic Grids

Sudoku and logic grids do not always reveal new hidden information after a move. Instead, a confirmed deduction propagates through the puzzle, shrinking the remaining candidate sets.

Computer science describes similar problems as constraint-satisfaction problems: variables have possible values, while rules restrict which combinations are allowed. One common strategy is to examine highly constrained variables first because contradictions or forced choices appear there sooner, as explained in Carnegie Mellon University’s constraint-satisfaction problem notes.

For a human solver, that means looking for:

• Cells with very few candidates
• Rows or groups with only a few open positions
• Clues containing words such as “exactly,” “immediately,” or “only”
• Locations where several restrictions overlap
• Decisions that would trigger multiple eliminations

The information-gain rule works naturally alongside the process of elimination: one helps you decide where to look, while the other tells you how to narrow the options.

Visual and Spatial Puzzles

In visual puzzles, information may come from testing orientation, fit, movement, color, or location.

A plain blue jigsaw piece might fit dozens of places. A piece containing part of a face, a letter, or a sharp color boundary is more informative because fewer locations can accept it. Similarly, in a sliding puzzle, a move that improves access to several trapped tiles can be more valuable than one that places a single tile correctly but blocks future movement.

Look for distinctive features and structural bottlenecks. These are the visual equivalents of highly constrained variables.

When Information Gain Is Not the Only Goal

The most informative move is not automatically the best move in every situation. Puzzle solving can involve competing priorities.

A guaranteed finishing move should usually beat an experiment. A timed puzzle may reward a quick, moderate-gain action over a slow calculation. A move that reveals a great deal but creates an irreversible mess may be too dangerous. You may also need to preserve resources such as hints, lives, blank tiles, open spaces, or remaining guesses.

Information gain should therefore guide judgment, not replace it. Balance it with:

• Certainty: Is the move logically proven?
• Progress: Does it advance the actual objective?
• Risk: What happens if the idea is wrong?
• Flexibility: Will future options remain open?
• Effort: Is the likely knowledge worth the time required?

Early in a puzzle, exploration often matters most. Near the end, certainty becomes more valuable because the remaining possibilities are already narrow.

Turn Every Move Into a Useful Question

Strong puzzle solvers are not people who instantly see every answer. They are people who repeatedly ask better questions.

Before your next move, pause and consider:

• What uncertainty am I trying to reduce?
• How many possibilities could this eliminate?
• Will every possible outcome teach me something?
• Does this move connect to other unresolved areas?
• Can I test it safely?
• Is there a more informative move elsewhere?

With practice, this pause becomes almost automatic. You stop treating a puzzle as a collection of isolated blanks and begin seeing it as a network of possibilities. Each move becomes a question, each result becomes evidence, and every useful piece of evidence brings the solution closer.