Solution: Why Your 100th Repetition Barely Feels Faster Than Your 90th

Numeric answer: T_16 = 3 seconds

T_16 = T_1 / √16 = 12 / 4 = 3

Part 2 — The shrinking-savings pattern

T_4   = 12 / √4   = 12 / 2  = 6.00
T_16  = 12 / √16  = 12 / 4  = 3.00
T_64  = 12 / √64  = 12 / 8  = 1.50
T_256 = 12 / √256 = 12 / 16 = 0.75

Absolute drops between consecutive 16×-practice steps:

T_4  → T_16:  6.00 − 3.00 = 3.00
T_16 → T_64:  3.00 − 1.50 = 1.50
T_64 → T_256: 1.50 − 0.75 = 0.75

Each 16×-more-practice step halves the absolute time saved by the previous step — 3.00, then 1.50, then 0.75. The relative improvement (each step still cuts time by the same factor, 4×, since √16 = 4) stays constant, but the absolute number of seconds gained keeps shrinking. This is exactly the diminishing-returns signature of a power law: proportional improvement per proportional increase in practice, but ever-smaller absolute payoff — which is why a beginner feels dramatic week-to-week gains while an expert’s improvement becomes nearly imperceptible trial-to-trial, even though (measured relatively) they may still be improving at a mathematically similar rate.

Part 3 — The floor the power law can’t cross

The power law T_N = T_1 × N^(−α) predicts T_N → 0 as N → ∞, which is physically impossible — no human task has zero execution time. Real practice curves flatten out and approach (but never cross) a nonzero asymptote, not literal zero. Concrete lower bounds, depending on the task:

• Motor/physical limits: nerve conduction velocity, muscle contraction speed, and biomechanical constraints set a hard floor on how fast any physical movement (a keystroke, a swing, a step) can physically execute, no matter how automatized the underlying skill becomes.
• Perceptual/processing limits: tasks requiring visual identification or decision-making are bounded by basic neural processing time — the minimum time for a stimulus to be perceived and acted on, which doesn’t shrink to zero regardless of practice.

The full, more accurate form researchers actually fit is often T_N = T_∞ + (T_1 − T_∞) × N^(−α) — an added asymptote term T_∞ (the floor) that the pure T_N = T_1 × N^(−α) form in this lesson simplified away for clean arithmetic. As N grows large, T_N approaches T_∞, not zero.

Part 4 — Power-law scaling and diminishing returns in AI training

If model performance improves as a power-law function of training compute (as neural scaling laws broadly describe), the same diminishing-returns logic applies directly: each additional fixed increment of performance improvement requires an ever-larger multiplicative increase in compute to purchase — going from “mediocre” to “good” might take 10× the compute of getting started, but going from “good” to “slightly better than good” might take another 10× on top of that for a much smaller absolute gain, mirroring Part 2’s shrinking absolute drops exactly.

Why this matters for real decisions: at some point on the curve, the compute cost of squeezing out the next small increment of performance via scale alone becomes so large that it stops being the most efficient lever — better/cleaner training data, an improved architecture, or an entirely different technique can potentially buy a comparable performance gain far more cheaply than another order-of-magnitude compute increase would. Recognizing “we’re on the flat part of a power-law curve” is exactly the signal that it’s time to look for a different lever, not just spend more on the same one — which is precisely the practical, resource-allocation reading of “diminishing returns” that Part 2’s arithmetic made visible.

The pattern

Rule: performance time falls off as a power function of practice — big early gains, shrinking later gains, approaching (never reaching) a physical/perceptual floor. It’s one of the most reliable quantitative laws in cognitive science, and it emerges directly from ACT-R’s activation mechanism rather than being a separate empirical patch on the architecture.

Where this goes: next lesson asks what’s actually changing inside the system as this curve plays out — expertise as chunking plus retrieval, using the classic de Groot chess studies that first revealed what separates a novice’s memory from a grandmaster’s.