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

TL;DR

This paper introduces two novel table-based encoding schemes, vectorized doomsdays and Doomyears, to simplify Conway's Doomsday Algorithm by replacing arithmetic with structured lookups, improving cognitive efficiency.

Contribution

The paper presents the Calamity Tables, a unified system of table encodings that replace arithmetic steps in the Doomsday Algorithm with lookup-based methods, with proofs and complexity analysis.

Findings

Proved correctness of the new encoding schemes.

Established self-verification properties of the encodings.

Compared cognitive complexity favorably against standard arithmetic methods.

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.