Sphere Quest Solver

v1.0 — 2026-04-07

Mode:

Click History

SP earned: 0 sp

Recommendation

Initializing…

State

Purple search

Arrangements left — Purples found 0 / 3 Clicks used 0 / 7

Cell Overlays — shown on unrevealed cells

	P% — chance of being purple Fraction of surviving arrangements that have a purple here. Starts at 16% everywhere; narrows quickly after each reveal. The recommended cell always has the highest P%.
	M — combined score (P + 0.1×G) The actual ranking value: P(purple) plus a small Gini bonus. Cells with equal P% are broken by how informative their reveal would be. Highest M cell = dashed recommendation.
	G — Gini impurity (informativeness) How mixed the possible reveal outcomes are. 0 = reveal is completely predictable (you’ll learn nothing new). Near 1 = outcome is uncertain, so whatever it shows will eliminate many arrangements. Center cells start high; cells boxed in by reveals tend to be low.
	EV — expected SP from this click Average SP value weighted across all possible reveal colors. Low during search (most cells are worth ~20–55 sp if not purple). Most meaningful in post-red phase when all purple positions are known and you’re spending remaining budget on the highest-value tiles.

Click any unrevealed cell to enter what color it revealed and the solver will recommend the best next click using a blend of the probability of being purple across all possible boards and the informational gain from that cell.

How the colors work

Each non-purple cell reveals how many of its up-to-8 neighbors are purple:

Blue — 0 purple neighbors
Teal — 1 purple neighbor
Green — 2 purple neighbors
Yellow — 3 purple neighbors
Orange — 4 purple neighbors

A blue cell rules out any purple in all 8 surrounding cells — very informative. An orange cell tells you exactly 4 of its 8 neighbors are purple, which strongly constrains where they can be. Purple clicks are free and instantly narrow the search.

How the probability of purple is calculated

Before any clicks, there are 12,650 equally likely ways to place 4 purples in 25 cells (that’s C(25,4)). Every color reveal eliminates the arrangements that contradict it — if you see a blue at (1,2), any arrangement that puts a purple adjacent to (1,2) is thrown out.

After each click, the simulator filters the list down to only the arrangements still consistent with everything revealed so far. P(purple) for a cell is then simply: how many of the surviving arrangements have a purple there, divided by the total surviving count. No guessing — exact counting.

On an empty board every cell starts at 16% (4 purples ÷ 25 cells). After a few reveals, some cells jump toward 100% and others drop to 0%.

Strategy comparison — exhaustive sim, all 12,650 boards

This solver uses the MIXED α=1.0 β=0.1 strategy, explained below.

Strategy	Avg SP	Success	Efficiency
Random	174.5	11.5%	45.7%
Mean field	190.3	24.2%	49.8%
Constraint prop	280.5	69.9%	73.4%
Exact P(purple)	296.8	81.1%	77.7%
MIXED α=1.0 β=0.1 ▶	342.7	95.4%	89.7%
Theoretical max	381.9	100%	100%

Success = fraction of boards where the red sphere was clicked.
Efficiency = avg score / avg theoretical max (381.9 sp).

How the strategies differ

Random — clicks any unrevealed cell at random. Finds red only 12% of the time because it wastes most of its budget on non-purple tiles.
Mean field — estimates purple probability from adjacent revealed cells independently, ignoring how constraints interact. Slightly better than random but still misses most boards.
Constraint propagation — minesweeper-style logic: if a cell says “1 purple neighbor” and only one neighbor is unknown, that neighbor must be purple. Reaches 70% success but stalls on ambiguous boards where no single constraint forces a deduction.
Exact P(purple) — maintains all surviving arrangements and picks the cell with the highest exact probability of being purple. Reaches 81% success but makes poor first clicks on an empty board where all cells look equal.
MIXED α=1.0, β=0.1 — ranks each cell by a combined score = α × P(purple) + β × Gini, where Gini is a representation of information gain that can help break ties when cells are close in P(purple). This helps to favor center cells on a fresh board and pushes success rate from 81% to 95%.

What is Gini impurity?

Gini impurity measures how “mixed” the possible outcomes of a click are. If a cell is very likely to be blue (based on all the surviving possible arrangements), that would be low impurity - you won't learn much from actually clicking it. If it could plausibly reveal a wide range of equally likely colors, that would be high impurity - whatever it shows, you can now eliminate a lof of different possibilities.

Concretely: take the probability of each possible outcome, square each one, sum them, subtract from 1. G = 1 − ∑ P(outcome)². Certain outcome → 0. Equally likely across 5 colors → near 1.

This is used here only as a tiebreaker (weight 0.1 vs. 1.0 for P%), so it only affects cases where two different cells have nearly equal purple probability.