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Using Simple Linear Models with Truncation to Determine the Gregorian Day of the Week

Bryce Iversen

TL;DR

The paper introduces a table-free algorithm to compute the day of the week in the Gregorian calendar by marrying a simple truncation-driven linear model with modulo arithmetic. Central to the method are the components $L(y)$ for leap-year accumulation, $D(y,m,d)$ for days into the year, and a final closed-form for $W(y,m,d)$ that yields the weekday after combining these terms with the epoch offset. Through two worked examples, the approach demonstrates correct weekday determination for dates spanning modern and recent history, while situating itself among classical formulas like Zeller’s congruence. The work emphasizes a minimal-memory, regression-based strategy to a perpetual calendar, with potential extensions to related calendar systems in future work.

Abstract

The Gregorian calendar -- first established for daily use on Friday, October 15th, 1582 by Pope Gregory XIII in Catholic countries -- is presently the most pervasive calendar in the world. As such, algorithms for performing various calendrical computations in accurate, performant, and easily implementable ways are extremely useful in fields like software engineering. In this paper, we present a novel algorithm for determining the day of the week for any date in the Gregorian calendar. Of note, our algorithm does not rely on remembering tables of values. Instead, we encode tables needed for computation using simple linear regression with truncation to adjust for any errors present in our linear models in such a way that no tables have to be recalled. In addition, our algorithm does not require a relabeling of days, weeks, months, or years to values other than their intuitive representations. The algorithm works by taking a date in the Gregorian calendar, calculating the number of days (accounting for leap years using simple linear regression with truncation) that have elapsed since the epoch of the Gregorian calendar in 1582 from the specified date and adding this number modulo 7 to the epoch's day of the week thus, obtaining the day of the week for the requested date in numeric form.

Using Simple Linear Models with Truncation to Determine the Gregorian Day of the Week

TL;DR

The paper introduces a table-free algorithm to compute the day of the week in the Gregorian calendar by marrying a simple truncation-driven linear model with modulo arithmetic. Central to the method are the components for leap-year accumulation, for days into the year, and a final closed-form for that yields the weekday after combining these terms with the epoch offset. Through two worked examples, the approach demonstrates correct weekday determination for dates spanning modern and recent history, while situating itself among classical formulas like Zeller’s congruence. The work emphasizes a minimal-memory, regression-based strategy to a perpetual calendar, with potential extensions to related calendar systems in future work.

Abstract

The Gregorian calendar -- first established for daily use on Friday, October 15th, 1582 by Pope Gregory XIII in Catholic countries -- is presently the most pervasive calendar in the world. As such, algorithms for performing various calendrical computations in accurate, performant, and easily implementable ways are extremely useful in fields like software engineering. In this paper, we present a novel algorithm for determining the day of the week for any date in the Gregorian calendar. Of note, our algorithm does not rely on remembering tables of values. Instead, we encode tables needed for computation using simple linear regression with truncation to adjust for any errors present in our linear models in such a way that no tables have to be recalled. In addition, our algorithm does not require a relabeling of days, weeks, months, or years to values other than their intuitive representations. The algorithm works by taking a date in the Gregorian calendar, calculating the number of days (accounting for leap years using simple linear regression with truncation) that have elapsed since the epoch of the Gregorian calendar in 1582 from the specified date and adding this number modulo 7 to the epoch's day of the week thus, obtaining the day of the week for the requested date in numeric form.

Paper Structure

This paper contains 12 sections, 17 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. The Formula
  4. Counting Leap Years
  5. Counting the number of days into a year
  6. Determining the Day of the Week
  7. Examples
  8. November 9, 1989 The Fall of the Berlin Wall
  9. July 26, 2024 The Paris 2024 Summer Olympics Opening Ceremony
  10. Discussion & Conclusion
  11. Future Work
  12. Acknowledgments

Figures (5)

  • Figure 1: A plot of the leap years since 1200 up until 1300 assuming that the Gregorian method of counting leap years was extended backwards in time
  • Figure 2: A plot of the leap years since 1200 from 1568 to 1632
  • Figure 3: A plot of the leap years since 1200 from 1668 to 1732
  • Figure 4: A plot of the leap years since 1200 from 1582 to 2150
  • Figure 5: A plot of the linear model predicting the day of the year (for a non-leap year) from the month