Each item in the bank follows the two-parameter logistic (2PL) IRT model: the probability an examinee of ability θ answers item i correctly is
P_i(θ) = 1 / (1 + exp(-a_i (θ - b_i)))
b_i = difficulty (θ where P = 0.5)
a_i = discrimination (steepness of the curve)
After every response the engine re-estimates θ by Newton–Raphson maximum likelihood over every item administered so far (u_i = 1 correct, 0 incorrect):
score: L'(θ) = Σ a_i (u_i - P_i(θ))
information: I(θ) = Σ a_i² P_i(θ)(1 - P_i(θ))
update: θ ← θ + L'(θ) / I(θ) (step damped to ±1, θ clamped to ±4)
SE(θ̂) ≈ 1 / √I(θ̂)
The Max information strategy — real computerized adaptive testing (CAT), as used by the GRE and many licensing exams — administers whichever remaining item maximizes I(θ) at the current estimate, since that item narrows the confidence interval the fastest. The Random strategy administers items in random order for comparison: watch SE(θ̂) shrink far more slowly for the same number of items.
This 2D version plots the mathematics directly instead of a spatial item bank: the top panel draws the actual item-characteristic curves P_i(θ) for every administered item (green = correct, red = incorrect) plus a faint sample of the unadministered pool, with a shaded band at θ̂ ± SE(θ̂). The middle panel plots the running log-likelihood ℓ(θ) = Σ[u·lnP + (1-u)·ln(1-P)] over the whole θ range — its peak is exactly the Newton–Raphson estimate, so you can see the MLE visually confirmed as the curve's maximum. The bottom panel traces θ̂ (with its ±SE band) after each administered item, so the narrowing of the interval under each strategy is directly comparable.