Leap year starting on Thursday
A leap year starting on Thursday is a year in the Gregorian calendar that contains 366 days, including February 29, and in which January 1 falls on a Thursday.[1][2] Such a year ends on December 31, which is a Friday, due to the extra day shifting the total of 366 days (equivalent to 52 weeks and 2 additional days) from the starting Thursday.[2] These years follow the standard Gregorian leap year rules, occurring every four years except for century years not divisible by 400, and their occurrence in the weekly cycle depends on the 400-year Gregorian cycle.[1] In the calendar for a leap year starting on Thursday, the dominical letters are DC, meaning the first part of the year (January to February 28) uses D for Sundays, shifting to C after the leap day. Notable features include two instances of Friday the 13th, occurring in February and August. Examples of such years in the 20th and 21st centuries include 1948, 1976, and 2004, with the next being 2032 and 2060.[2][3][4] The structure affects the placement of holidays and observances; for instance, New Year's Day is a Thursday, Valentine's Day falls on a Saturday, and Christmas on a Saturday. These years contribute to the overall variability in the Gregorian calendar's alignment with the seven-day week, repeating patterns irregularly every 6, 11, or 28 years depending on intervening leap years and century rules.[1]Calendar Fundamentals
Definition and Occurrence
A leap year in the Gregorian calendar is a year with 366 days, achieved by adding an extra day, February 29, to align the calendar more closely with the solar year of approximately 365.2425 days.[5] This occurs for years divisible by 4, except for century years (divisible by 100), which are not leap years unless also divisible by 400; thus, years like 1700, 1800, and 1900 are common years, while 2000 is a leap year.[6] The Gregorian calendar, introduced in 1582, uses this rule to maintain seasonal consistency over long periods.[7] A leap year starting on Thursday specifically refers to one where January 1 falls on a Thursday, resulting in 366 days and ending on Friday, December 31. This configuration produces distinct weekday alignments across the months; for instance, February 1 falls on a Sunday, and the leap day shifts subsequent dates such that March 1 lands on a Monday.[8] Such alignments affect the distribution of weekdays for events and observances throughout the year. These leap years occur once within each 28-year solar cycle of the Gregorian calendar, a period encompassing 28 common years and 7 leap years where weekday patterns repeat due to the combined effects of 365-day years advancing the calendar by 1 weekday and leap years by 2.[9] The pattern generally repeats every 28 years, but century year exceptions—skipping leap days in non-400-divisible centuries—can interrupt this cycle, altering the frequency over longer spans like the full 400-year Gregorian cycle.[5] Mathematically, the starting weekday of a leap year can be determined using Zeller's Congruence, an algorithm for computing the day of the week for any Gregorian date:h \equiv q + \left\lfloor \frac{13(m+1)}{5} \right\rfloor + K + \left\lfloor \frac{K}{4} \right\rfloor + \left\lfloor \frac{J}{4} \right\rfloor - 2J \pmod{7}
where h = 0 for Saturday, 1 for Sunday, ..., 6 for Friday; q is the day of the month; m is the month (March = 1, April = 2, ..., February = 12 of the prior year); K is the year of the century (year \mod 100); and J is the century (\lfloor year / 100 \rfloor). For January 1 in a leap year, treat it as day 1 of month 13 in the previous year to account for the leap day shift after February, ensuring the formula incorporates the extra day correctly.[8] The Doomsday rule provides a mnemonic alternative for mental verification of such dates.[10]
Doomsday Rule Application
The Doomsday rule, developed by mathematician John Horton Conway, enables the determination of any Gregorian calendar date's weekday by anchoring it to the year's "doomsday"—the shared weekday for specific memorable dates within each month, such as 4/4 for April, 6/6 for June, 8/8 for August, 10/10 for October, and 12/12 for December.[11] In leap years starting on Thursday, the doomsday falls on Sunday, meaning these key dates occur on Sundays throughout the year.[12] Leap years require specific adjustments to the memorable dates for January and February to account for the additional day on February 29, which shifts the calendar after February. January's doomsdays are the 4th, 11th, 18th, or 25th, while February's are the 29th, 8th, 15th, or 22nd (with February 28 or 29 serving as a primary reference, equivalent to "March 0").[13] The year's doomsday can be computed via the formula anchor = (2 + 5 × (year mod 4) + 4 × (year mod 100) + 6 × (year mod 400)) mod 7, where 0 denotes Sunday, 1 Monday, and so on up to 6 for Saturday; for leap years starting on Thursday, this evaluates to 0 (Sunday).[14] To calculate a date's weekday, identify the closest doomsday date for the month, then add or subtract the day difference modulo 7 from the known doomsday (Sunday). Month doomsdays follow these patterns: even months (April–December) use the month number as the date; March uses 7, 14, 21, or 28; odd months after March use 9/5 (September), 7/11 (July), 11/7 (November). For instance, in such a leap year, March 7 (a doomsday) is Sunday, so March 14 (another doomsday, 7 days later) is also Sunday, and even months like August 8 fall on Sunday. To find September 9, note September's doomsday is the 5th (Sunday); add 4 days: September 6 Monday, 7 Tuesday, 8 Wednesday, 9 Thursday—thus, Thursday.[11][15] Mnemonics facilitate recall: even-month dates rhyme with the month ("4/4, 6/6, 8/8"), while odd months use "9-5 at the 7-11" for September 5, July 11, and November 7. For leap-year January and February, remember "January 4" or "February 29/last day of February." Since the year starts on Thursday, a quick check confirms the doomsday: January 1 Thursday, 2 Friday, 3 Saturday, 4 Sunday—aligning with the formula's result.[16][13]Historical and Applicable Years
Gregorian Calendar Examples
In the Gregorian calendar, which follows a 400-year cycle comprising 97 leap years, leap years beginning on Thursday exhibit a recurring pattern influenced by the 28-year solar cycle, during which the calendar's weekday alignment typically repeats due to the accumulation of 365 days (1 extra day mod 7) plus 7 leap days (another full week). However, this pattern is interrupted at century years not divisible by 400 (e.g., 1700, 1800, 1900), which omit a leap day and cause an additional shift in the weekday progression.[5][11] Historical examples in the modern Gregorian calendar (post-1582 adoption) include 1756, 1784, 1824, 1852, 1880, 1920, 1948, 1976, and 2004, each verified to start on Thursday as leap years.[22][23][24][25] Notably, the gap between 1880 and 1920 spans 40 years rather than 28, owing to 1900's exclusion as a leap year.[5] Future projections follow similar intervals: after 2004, the next occurrences are 2032, 2060, and 2088, with a 40-year skip to 2128 due to 2100 not being a leap year.[26][27][28][29][5] The doomsday rule can verify these alignments, where such leap years have a doomsday (reference weekday for key dates) of Sunday.[11] In the proleptic Gregorian calendar—extending the rules backward before 1582—examples include 1604 and 1632, both leap years starting on Thursday, illustrating the pattern's consistency prior to the reform's implementation.Julian Calendar Examples
The Julian calendar applies a simple leap year rule, inserting an extra day in February every fourth year without exceptions for century years, resulting in an average year length of 365.25 days. This contrasts with the Gregorian calendar by lacking the century rule exceptions, causing the Julian calendar to drift ahead of the solar year by about one day every 128 years.[5] The Julian calendar's weekday patterns repeat every 28 years, as this period contains 10,227 days—precisely divisible by 7—encompassing 21 ordinary years and 7 leap years. Within this cycle, leap years starting on Thursday occur at regular intervals, specifically every 28 years, corresponding to years congruent to 0 modulo 28 in the Julian reckoning. The doomsday rule facilitates identification of these years: the doomsday (the weekday of key anchor dates like February 29 in leap years) is calculated as (6 × C + Y + ⌊Y/4⌋) mod 7, where the year is 100C + Y; for leap years starting on Thursday, this yields a doomsday of 0 (assuming Sunday = 0, Monday = 1, ..., Saturday = 6).[32] Historical examples of leap years starting on Thursday in the Julian calendar include 28 AD, 56 AD, 84 AD, 112 AD, 140 AD, 196 AD, 252 AD, 308 AD, 364 AD, 420 AD, 476 AD, 532 AD, 588 AD, 644 AD, 700 AD, 756 AD, 812 AD, 868 AD, 924 AD, 980 AD, 1036 AD, 1092 AD, 1148 AD, 1204 AD, 1260 AD, 1316 AD, 1372 AD, 1428 AD, 1484 AD, 1540 AD, and 1568 AD, spanning from the early Roman Empire through the late Middle Ages up to the widespread adoption of the Gregorian calendar in 1582. These years illustrate the consistent pattern before the transition in Western Europe.[32] The Julian calendar remained in use for ecclesiastical purposes in the Eastern Orthodox Church, with some regions adhering to it until the early 20th century, such as Russia until 1918 and Greece until 1923. This extended usage meant ongoing application of the every-four-years leap rule, including in years like 1900, which was a leap year in the Julian system but started on Sunday according to verified calculations. The absence of Gregorian adjustments thus perpetuated the drift, affecting holiday alignments in Orthodox traditions.Holiday and Observance Impacts
International and Fixed-Date Holidays
In leap years starting on Thursday in the Gregorian calendar, fixed-date international holidays observe specific weekdays determined by the year's structure, with the insertion of February 29 causing a one-day shift for all dates from March onward compared to non-leap years beginning on the same day.[11] New Year's Day, celebrated globally on January 1, always falls on a Thursday, marking the start of the year.[33] February 29, the leap day observed in some countries as a public holiday or special occasion, occurs on a Sunday; this follows from January's 31 days advancing Thursday by 3 weekdays to make February 1 a Sunday, with the subsequent 28 days to February 29 equaling exactly 4 weeks and preserving the Sunday.[34] The extra leap day shifts March 1 to a Monday, altering the weekday pattern for subsequent months and affecting fixed-date observances. For instance, International Workers' Day on May 1 falls on a Saturday, providing a weekend alignment in many regions.[35] United Nations Day, held on October 24, lands on a Sunday, often extending observances into the following week.[36] Similarly, fixed-date events like Bastille Day on July 14 occur on a Wednesday, influencing mid-week commemorations in participating countries.[37] This post-February shift by one weekday—derived from the Doomsday rule's application to leap years—can impact global coordination for holidays, such as adjusting travel or event scheduling around weekends versus weekdays.[11]Regional and Movable Observances
In Australia and New Zealand, leap years beginning on a Thursday result in Waitangi Day, commemorated on February 6, falling on a Friday.[38] Similarly, ANZAC Day on April 25 occurs on a Sunday, as seen in 2004 when the date aligned with weekend observances across both nations.[39] In Canada, the fixed-date Canada Day on July 1 lands on a Thursday in these years, influencing public gatherings and fireworks displays.[40] For movable holidays, Victoria Day—the Monday preceding May 25—is affected by the year's weekday progression; with May 25 falling on a Tuesday, the holiday shifts to May 24, a Monday, altering the timing of the long weekend compared to non-leap years starting on the same day. Across the British Isles, Boxing Day on December 26 falls on a Sunday, leading to the bank holiday being observed on the following Monday (December 27), though the overall yearly pattern impacts adjacent bank holidays that are routinely adjusted to Mondays for extended breaks.[41] In Denmark, Constitution Day on June 5 occurs on a Saturday, often leading to combined weekend celebrations.[42] In Germany, German Unity Day on October 3 aligns with a Sunday, prompting observances that may extend into the following Monday in some regions.[43] Movable observances like Easter, determined by the lunar calendar, remain Sundays independent of the year's starting weekday, though in leap years beginning on Thursday—such as 2004 when Easter fell on April 11—the alignment can position it closer to midweek weekdays for preceding events like Good Friday. Practical implications include school holidays and election schedules in these regions, which tie to specific weekdays and may start or end on Fridays or Mondays due to the fixed pattern, affecting attendance and logistics.[25]Religious Solemnities
In leap years commencing on a Thursday in the Gregorian calendar, such as 2004, the Roman Catholic liturgical calendar experiences variable placements for movable feasts like Ash Wednesday due to the computus algorithm determining Easter's date. For instance, in 2004, Ash Wednesday fell on February 25, a Wednesday, marking the start of Lent 46 days before Easter Sunday on April 11.[44] Fixed-date solemnities in these years align with consistent weekdays based on the calendar's structure. The Feast of Saint Patrick on March 17 occurs on a Wednesday, as it did in 2004, emphasizing themes of evangelism and national patronage in Irish Catholic tradition. Similarly, All Saints' Day on November 1 falls on a Monday, providing a solemn observance of the communion of saints immediately following All Souls' Day.[45][46] The insertion of February 29, which falls on a Sunday in such years, historically influenced the placement of certain saints' feasts in the pre-1969 Roman Calendar, particularly that of Saint Matthias the Apostle. Traditionally assigned to the sixth day before the Calends of March (February 24 in non-leap years), the feast shifted to February 25 in leap years to maintain the vigil on the preceding day, accommodating the duplicated dating structure before the leap day; this adjustment, known as the "leaping saint" tradition, ensured liturgical continuity until the 1969 calendar revision moved it permanently to May 14.[47][48] The liturgical year in the Roman Rite begins with the First Sunday of Advent, calculated as the Sunday nearest to November 30 (falling between November 27 and December 3), which in these leap years aligns with November 28, a Sunday, as seen in 2004; this initiates the Advent season of preparation, spanning four Sundays until Christmas.[49][50] In Eastern Orthodox traditions, which compute Pascha (Easter) using the Julian calendar, leap years in the Gregorian sense like 2032 result in distinct alignments, with Pascha observed on May 2, a Sunday, diverging from Western Easter on March 28 due to the 13-day Julian lag and metonic cycle differences.[51]Cultural and Practical Significance
Weekday Patterns for Events
In a leap year beginning on Thursday, the distribution of weekdays across the 366 days follows a specific sequence determined by the Gregorian calendar's month lengths and the extra day in February. This pattern is consistent for all such years and aids in scheduling events that recur on particular weekdays, such as weekly meetings or seasonal activities spanning multiple months. The year advances the calendar by two weekdays compared to the previous non-leap year, resulting in 52 full weeks plus two extra days.[25] January spans from Thursday to Saturday, featuring five Thursdays, five Fridays, and five Saturdays, with four of each other weekday. February runs from Sunday to Sunday, including the leap day on Sunday, and contains five Sundays along with four of each remaining weekday. March extends from Monday to Wednesday, with five Mondays, five Tuesdays, and five Wednesdays, and four of the others. April covers Thursday to Friday, offering five Thursdays and five Fridays, and four others. May goes from Saturday to Monday, including five Saturdays, five Sundays, and five Mondays. June proceeds from Tuesday to Wednesday, with five Tuesdays and five Wednesdays. July mirrors January, starting Thursday and ending Saturday, with five Thursdays, Fridays, and Saturdays. August aligns with February and December in starting on Sunday but ends on Tuesday, featuring five Sundays, five Mondays, and five Tuesdays. September starts Wednesday and ends Thursday, with five Wednesdays and five Thursdays. October runs Friday to Sunday, including five Fridays, five Saturdays, and five Sundays. November starts Monday and ends Tuesday, with five Mondays and five Tuesdays. December begins Wednesday and ends Friday, featuring five Wednesdays, Thursdays, and Fridays. These distributions can be verified using the Doomsday rule, where the anchor day for the year is Sunday.[52][53][54] Quarterly patterns in such years show the first quarter concluding on a Wednesday (March 31), the second on a Wednesday (June 30), the third on a Thursday (September 30), and the fourth on a Friday (December 31). This sequence influences fiscal reporting and business cycles that align with quarter ends, as the mid-year quarters fall mid-week, potentially affecting operational planning.[25] For event planning, these leap years correspond to ISO 8601 years with 53 weeks, occurring because the year starts on a Thursday; this extends the week count beyond the standard 52, impacting systems like payroll or project timelines that use ISO weeks. The overall structure of 52 weeks plus two days means the next year begins two weekdays later than the prior non-leap year would suggest.[55]| Key Date | Weekday |
|---|---|
| April 1 | Thursday |
| July 4 | Sunday |
| December 25 | Saturday |