The science
Rain to runoff (SCS curve number). Soil and cover store some rain before any runs off. With potential retention S = 25400/CN − 254 (mm) and initial abstraction Ia = 0.2 S, the cumulative direct runoff is Q = (P − Ia)2/(P − Ia + S) for P > Ia. A higher CN (paved, saturated) means less storage and more runoff. The curve number is an empirical method, not a physical law; 0.2 is a convention.
Runoff to hydrograph (SCS unit hydrograph). Excess rain, step by step, is spread over time by a triangular unit hydrograph: time to peak tp = D/2 + 0.6 Tc, base tb = 2.67 tp, peak qp = 0.208 A / tp m³/s per mm (A in km², t in h). Its area equals exactly 1 mm over the catchment; the hydrograph is the sum of these triangles for each time step (linear superposition).
Flow to stage (Manning). Q = (1/n) A R2/3 √S. The valley section is divided into main channel (trapezoid, 12 m wide, 1:2 banks, 2 m deep) and floodplain; each part has its own roughness and the flows add (divided-channel method). The stage is the root of Q(h) = Qin found by bisection. Water above bankfull spills across the floodplain, where it is slower (larger n) and wider, so stage rises more slowly with discharge.
Houses. A house is counted as flooded when the river stage exceeds its ground level plus a 0.2 m plinth. Damage is not modelled; house positions are illustrative.
Simplifications: uniform steady flow at each instant (no flood wave travel, no storage attenuation, no backwater), one prismatic reach, a uniform stage across the valley, uniform rain over the catchment, no snowmelt, no infiltration recovery between bursts, a triangular unit hydrograph in place of a measured one, a fixed 0.2 abstraction ratio. Real flood forecasting uses calibrated models and gauges; this is a teaching chain.
Independent educational model; the valley and houses are stylised illustrations.