The sample space
Eight fair independent bits produce 2⁸ = 256 equally likely ordered words. Any specified complete word therefore has probability 1/256. Knowing an already revealed prefix leaves each unseen bit fair in the model; previous rounds do not change the distribution of the next draw.
The absence test
For each length k in {2,3,4}, enumerate all 2ᵏ binary strings p. Include p exactly when the word does not contain p but contains both p without its last bit and p without its first bit. Occurrences are contiguous and may overlap. Sum k over included strings; do not count occurrences separately.
Examples you can inspect
00000000 and 11111111 both have raw score zero. For 01010101 the qualifying patterns are 00 and 11, yielding score four. Its gross multiplier in linear mode is 0.96 × 4 / 14.171875, approximately 0.2709592. Gross includes the returned amount after the stake has already been deducted.
Normalization and net change
Let S be raw score and m be 1, 2 or 3. Gross equals stake × 0.96 × Sᵐ / E[Sᵐ]. Exact means are 14.171875, 240.34375 and 4584.34375. Taking the expectation gives 0.96 × stake; expected net change is therefore −0.04 × stake before display rounding. Fractional carry preserves returns between rounds.
Zero and conditional outcomes
Every mixed word contains both 0 and 1; at least one minimal absent pattern of length two through four exists in this finite sample space. Exact enumeration finds only the two constant words score zero: 2/256 = 0.78125%. Raising a zero score to another positive power leaves it zero, so all modes share that probability.
Timing is not a strategy
The word is captured when the stake is deducted and settles after 480 milliseconds. Stop only removes queued future draws; it does not select a different word. Neither the delay nor a sound toggle changes the scoring function. A 96% average over the full sample space does not promise recovery after a losing sequence.
Questions & answers
What makes a pattern qualify?
It must be absent while both its prefix and suffix of one shorter length occur contiguously in the word.
Does cubed mode improve expected return?
No. Each mode is normalized separately to 96% expected gross return. The outcome distribution changes.
Does a loss make a win more likely next?
No. Each new eight-bit word is independently sampled with the same distribution.
