Table-Based Encodings for Conway's Doomsday Algorithm: Vectorized Doomsdays and Doomyears

Abstract

Conway's Doomsday Algorithm (1973) determines the day of the week for any date in the Gregorian calendar via three additive components: a century anchor, a year offset, and a month-day offset. The century anchor is a fixed four-entry table. The other two components require live arithmetic: the year offset demands computing $y + \lfloor y/4 \rfloor \pmod{7}$, and the month-day offset requires a subtraction that can produce negative intermediate values. We present two new encoding schemes that replace both arithmetic steps with structured table lookups. The first, vectorized doomsdays, re-encodes each month's doomsday date as a two-digit number whose tens and units digits represent the backward and forward gaps (respectively) from the nearest multiples-of-seven month anchors. A directional crossing rule (the "square knot rule") pairs the target date's gap with the opposite-direction digit, reducing the month-day offset to a single-digit addition. The second, Doomyears, encodes the year-offset function as a navigational lookup exploiting the 28-year periodicity of the Gregorian weekday cycle. Together with Conway's century anchor table, these form a unified system we call the Calamity Tables. We prove correctness, establish self-verification properties, analyse the internal structure of both encodings, and compare the cognitive complexity of the Calamity Table system against the standard arithmetic method.