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Roman calendar
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The Roman calendar is the calendar used by the Roman kingdom and republic. It is often inclusive of the Julian calendarestablished by the reforms of the dictator Julius Caesar and emperor Augustus in the late 1st century bc and sometimes inclusive of any system dated by inclusive counting towards months' kalends, nones, and ides in the Roman manner. It is usually exclusive of the Alexandrian calendar of Roman Egypt, which continued the unique months of that land's former calendar; the Byzantine calendar of the later Roman Empire, which usually dated the Roman months in the simple count of the ancient Greek calendars; and the Gregorian calendar, which refined the Julian system to bring it into still closer alignment with the solar year and is the basis of the current international standard.
Roman dates were counted inclusively forward to the next of three principal days: the first of the month (the kalends), a day less than the middle of the month (the ides), and eight days—nine, counting inclusively—before this (the nones). The original calendar consisted of 10 months beginning in spring with March; winter was left as an unassigned span of days. These months ran for 38 nundinal cycles, each forming a kind of eight (i.e., "nine") day week ended by religious rituals and a public market. The winter period was then used to create January and February. The legendary early kings Romulus and Numa were traditionally credited with establishing this early fixed calendar, which bears traces of its origin as an observational lunar one. In particular, the kalends, nones, and ides seem to have derived from the first sighting of the crescent moon, the first-quarter moon, and the full moon respectively. The system ran well short of the solar year, and it needed constant intercalation to keep religious festivals and other activities in their proper seasons. For superstitious reasons, such intercalation occurred within the month of February even after it was no longer considered the last month.
After the establishment of the Roman Republic, years began to be dated by consulships and control over intercalation was granted to the pontifices, who eventually abused their power by lengthening years controlled by their political allies and shortening the years in their rivals' terms of office. Having won his war with Pompey, Caesar used his position as Rome's chief pontiff to enact a calendar reform in 46 bc, coincidentally making the year of his third consulship last for 446 days. In order to avoid interfering with Rome's religious ceremonies, the reform added all its days towards the ends of months and did not adjust any nones or ides, even in months which came to have 31 days. The Julian calendar was supposed to have a single leap day on 24 February (a doubled VI Kal. Mart.) every fourth year but following Caesar's assassination the priests figured this using inclusive counting and mistakenly added the bissextile day every three years. In order to bring the calendar back to its proper place, Augustus was obliged to suspend intercalation for a few decades. The revised calendar remaining slightly longer than the solar year, the date of Easter shifted far enough away from the vernal equinox that Pope Gregory XIII ordered its adjustment in the 16th century.
Contents
[hide]History[edit]
Prehistoric lunar calendar[edit]
The original Roman calendar is believed to have been an observational lunar calendar[1] whose months began from the first signs of a new crescent moon. Because a lunar cycle is about 291⁄2 days long, such months would have varied between 29 and 30 days. Twelve such months would have fallen 10 or 11 days short of the solar year; without adjustment, such a year would have quickly rotated out of alignment with the seasons in the manner of the present-day Islamic calendar. Given the seasonal aspects of the later calendar and its associated religious festivals, this was presumably avoided through some form of intercalation or through the suspension of the calendar during winter.
Rome's 8-day week, the nundinal cycle, was shared with the Etruscans, who used it as the schedule of royal audiences. It was presumably a feature of the early calendar and was credited in Roman legend variously to Romulus and Servius Tullius.
Legendary 10-month calendar[edit]
The Romans themselves described their first organized year as one with ten fixed months, each of 30 or 31 days.[2][3] Such a decimal division fit general Roman practice.[4] The four 31-day months were called "full" (pleni) and the others "hollow" (cavi).[a][6] Its 304 days made up exactly 38 nundinal cycles. The system is usually said to have left the remaining 50-odd days of the year as an unorganized "winter", although Licinius Macer's lost history apparently stated the earliest Roman calendar employed intercalation instead[7][8] and Macrobius claims the 10-month calendar was allowed to shift until the summer and winter months were completely misplaced, at which time additional days belonging to no month were simply inserted into the calendar until it seemed things were restored to their proper place.[9][10]
Later Roman writers credited this calendar to Romulus,[11][12] their legendary first king and culture hero, although this was common with other practices and traditions whose origin had been lost to them. Some scholars doubt the existence of this calendar at all, as it is only attested in late Republican and Imperial sources and apparently supported only by the misplaced names of the months from September to December.[13] Rüpke also finds the coincidence of the length of the supposed "Romulan" year with the length of the first ten months of the Julian calendar to be suspicious.[clarification needed][13]
| English | Latin | Meaning | Length in days [2][3] |
|---|---|---|---|
| March | Mensis Martius | Month of Mars | 31 |
| April | Mensis Aprilis | Uncertain | 30 |
| May | Mensis Maius | Uncertain | 31 |
| June | Mensis Iunius | Month of Juno | 30 |
| Quintilis | Mensis Quintilis Mensis Quinctilis[14] | Fifth Month | 31 |
| Sextilis | Mensis Sextilis | Sixth Month | 30 |
| September | Mensis September | Seventh Month | 30 |
| October | Mensis October | Eighth Month | 31 |
| November | Mensis November | Ninth Month | 30 |
| December | Mensis December | Tenth Month | 30 |
—
| (51) | ||
Other traditions existed alongside this one, however. Plutarch's Parallel Lives recounts that Romulus's calendar had been solar but adhered to the general principle that the year should last for 360 days. Months were employed secondarily and haphazardly, with some counted as 20 days and others as 35 or more.[15][16]
Republican calendar[edit]
The attested calendar of the Roman Republic was quite different. It followed Greek calendars in assuming a lunar cycle of 291⁄2 days and a solar year of 121⁄2 synodic months( 3683⁄4 days), which align every fourth year after the addition of two intercalary months.[6] The additional two months of the year were January and February; the intercalary month was sometimes known as Mercedonius.[6]
The Romans did not follow the usual Greek practice in alternating 29- and 30-day months and a 29- or 30-day intercalary month every other year. Instead, their 3rd, 5th, 7th, and 10th months[b] had 31 days each; all the other months had 29 days except February, which had 28 days for three years and then 29 every fourth year. The total of these months over a 4-year span differed from the Greeks by 5 days, meaning the Roman intercalary month always had 27 days. Similarly, within each month, the weeks did not vary in the Greek fashion between 7 and 8 days; instead, the full months had two additional days in their first week and the other three weeks of every month ran for 8 days ("nine" by Roman reckoning).[17] Still more unusually, the intercalary month was not placed at the end of the year but within the month of February after the Terminalia on the 23rd (a.d. VIIKal. Mart.); the remaining days of February followed its completion. This seems to have arisen from Roman superstitions concerning the numbering and order of the months.[citation needed] The arrangement of the Roman calendar similarly seems to have arisen from Pythagorean superstitions concerning the luckiness of odd numbers.[17]
These Pythagorean-based changes to the Roman calendar were generally credited by the Romans to Numa Pompilius, Romulus's successor and the second of Rome's seven kings,[citation needed] as were the two new months of the calendar.[18][19][c] Most sources thought he had established intercalation with the rest of his calendar.[citation needed]Although Livy's Numa instituted a lunar calendar, the author claimed the king had instituted a 19-year system of intercalation equivalent to the Metonic Cycle[20] centuries before its development by Babylonian and Greek astronomers.[d] Plutarch's account claims he ended the former chaos of the calendar by employing 12 months totaling 354 days—the length of the lunar and Greek years—and biennial intercalary months of 22 days.[15][16]
Plutarch believed Numa was responsible for placing January and February first in the calendar;[15][16] Ovid states January began as the first month and February the last, with its present order owing to the Decemvirs.[22][23] W. Warde Fowler believed the Roman priests continued to treat January and February as the last months of the calendar throughout the Republican period.[24]
| English | Latin | Meaning | Length in days [25][26][15][16] |
|---|---|---|---|
| January | Mensis Ianuarius | Month of Janus | 29 |
| February | Mensis Februarius | Month of the Februa | 28 |
| Mercedonius Intercalary Month | Mercedonius Mensis Intercalaris | Month of Wages | 23 |
| March | Mensis Martius | Month of Mars | 31 |
| April | Mensis Aprilis | Uncertain | 29 |
| May | Mensis Maius | Uncertain | 31 |
| June | Mensis Iunius | Month of Juno | 29 |
| Quintilis | Mensis Quintilis Mensis Quinctilis[14] | Fifth Month | 31 |
| Sextilis | Mensis Sextilis | Sixth Month | 29 |
| September | Mensis September | Seventh Month | 29 |
| October | Mensis October | Eighth Month | 31 |
| November | Mensis November | Ninth Month | 29 |
| December | Mensis December | Tenth Month | 29 |
The consuls' terms of office were not always a modern calendar year, but ordinary consuls were elected or appointed annually. The traditional list of Roman consuls used by the Romans to date their years began in 509 bc.[27]
Flavian reform[edit]
Gnaeus Flavius, a secretary to the pontifex maximus, introduced a series of reforms in 304 bc.[28] Their exact nature is uncertain, although he is thought to have begun the custom of publishing the calendar in advance of the month, depriving the priests of some of their power but allowing for a more consistent calendar for official business.[29]
Julian reform[edit]
Julius Caesar, following his victory in his civil war and in his role as pontifex maximus, ordered a reformation of the calendar in 46 bc. This was undertaken by a group of scholars apparently including the Alexandrian Sosigenes[30] and the Roman M. Flavius.[31][26] Its main lines involved the insertion of ten additional days throughout the calendar and regular intercalation of a single leap day every fourth year in order to bring the Roman calendar into close agreement with the solar year. The year 46 bc was the last of the old system and included 3 intercalary months, the first inserted in February and two more—Intercalaris Prior and Posterior—before the kalends of December.
Later reforms[edit]
After Caesar's assassination, Mark Antony had his birth month Quintilis renamed July (Iulius) in his honor. After Antony's defeat at Actium, Augustus assumed control of Rome and, finding the priests had (owing to their inclusive counting) been intercalating every third year instead of every fourth, suspended the addition of leap days to the calendar for a number of decades until its proper position had been restored. In 8 bc, the plebiscite Lex Pacuvia de Mense Augusto caused Sextilis to be renamed August (Augustus) in his honor.[32][33][26][e]
In large part, this calendar continued unchanged under the Roman Empire. (Egyptians used the related Alexandrian calendar, which Augustus had adapted from their wandering ancient calendar to maintain its alignment with Rome's.) A few emperors altered the names of the months after themselves or their family, but such changes were abandoned by their successors. Diocletian began the 15-year indiction cycles beginning from the ad 297 census;[27] these became the required format for official dating under Justinian. Constantine formally established the 7-day week by making Sunday an official holiday in 321. Consular dating became obsolete following the abandonment of appointing nonimperial consuls in ad 541.[27] The Roman method of numbering the days of the month never became widespread in the Hellenized eastern provinces and was eventually abandoned by the Byzantine Empire in its calendar.
Days[edit]
Roman dates were counted inclusively forward to the next one of three principal days within each month:[34]
- Kalends (Kalendae or Kal.), the 1st day of each month[34]
- Nones (Nonae or Non.), the 7th day of full months[35] and 5th day of hollow ones,[34] 8 days—"nine" by Roman reckoning—before the Ides in every month
- Ides (Idus, variously Eid. or Id.), the 15th day of full months[35] and the 13th day of hollow ones,[34] a day less than the middle of each month
These are thought to reflect a prehistoric lunar calendar, with the kalends proclaimed after the sighting of the first sliver of the new crescent moon a day or two after the new moon, the nones occurring on the day of the first-quarter moon, and the ides on the day of the full moon. The kalends of each month were sacred to Juno and the ides to Jupiter.[36][37] The day before each was known as its eve (pridie); the day after each (postridie) was considered particularly unlucky.
The days of the month were expressed in early Latin using the ablative of time, denoting points in time, in the contracted form "the 6th December Kalends" (VI Kalendas Decembres).[35] In classical Latin, this use continued for the three principal days of the month[38] but other days were idiomatically expressed in the accusative case, which usually expressed a duration of time, and took the form "6th day before the December Kalends" (ante diem VI Kalendas Decembres). This anomaly may have followed the treatment of days in Greek,[39] reflected the increasing use of such date phrases as an absolute phrase able to function as the object of another preposition,[35] or simply originated in a mistaken agreement of dies with the preposition ante once it moved to the beginning of the expression.[35] In late Latin, this idiom was sometimes abandoned in favor of again using the ablative of time.
The kalends were the day for payment of debts and the account books (kalendaria) kept for them gave English its word calendar. The public Roman calendars were the fasti, which designated the religious and legal character of each month's days. The Romans marked each day of such calendars with the letters:[40]
- F (fastus, "permissible") on days when it was legal to initiate action in the courts of civil law (dies fasti, "allowed days")
- C (comitialis) on fasti days during which the Roman people could hold assemblies (dies comitalis)
- N (nefastus) on days when political and judicial activities were prohibited (dies nefasti)
- NP (uncertain)[f] on public holidays (feriae)
- QRCF (uncertain)[g] on days when the "king" (rex sacrorum) could convene an assembly
- EN (endotercissus, an archaic form of intercissus, "halved") on days when most political and religious activities were prohibited in the morning and evening due to sacrificesbeing prepared or offered but were acceptable for a period in the middle of the day
Each day was also marked by a letter from A to H to indicate its place within the nundinal cycle of market days.
Weeks[edit]
The nundinae were the market days which formed a kind of weekend in Rome, Italy, and some other parts of Roman territory. By Roman inclusive counting, they were reckoned as "ninth days" although they actually occurred every eighth day. Because the republican and Julian years were not evenly divisible into eight-day periods, Roman calendars included a column giving every day of the year a nundinal letter from A to H marking its place in the cycle of market days. Each year, the letter used for the markets would shift 2–5 letters along the cycle. As a day when the city swelled with rural plebeians, they were overseen by the aediles and took on an important role in Roman legislation, which was supposed to be announced for three nundinal weeks (between 17 and 24 days) in advance of its coming to a vote. The patricians and their clients sometimes exploited this fact as a kind of filibuster, since the tribunes of the plebs were required to wait another three-week period if their proposals could not receive a vote before dusk on the day they were introduced. Superstitions arose concerning the bad luck that followed a nundinae on the nones of a month or, later, on the first day of January. Intercalation was supposedly used to avoid such coincidences, even after the Julian reform of the calendar.
The 7-day week began to be observed Italy in the early imperial period,[42] as practitioners and converts to eastern religions introduced Hellenistic and Babylonian astrology, the Jewish Saturday sabbath, and the Christian Lord's Day. The system was originally used for private worship and astrology but had replaced the nundinal week by the time Constantine made Sunday (dies Solis) an official day of rest in ad 321. The hebdomadal week was also reckoned as a cycle of letters from A to G; these were adapted for Christian use as the dominical letters.
Months[edit]
The names of Roman months originally functioned as adjectives (e.g., the January kalends occur in the January month) before being treated as substantive nouns in their own right (e.g., the kalends of January occur in January). Some of their etymologies are well-established: January and March honor the gods Janus[43] and Mars;[44] July and August honor the dictator Julius Caesar[45] and his successor, the emperor Augustus;[46] and the months Quintilis,[47] Sextilis,[48] September,[49] October,[50] November,[51] and December[52] are archaic adjectives formed from the ordinal numbers from 5 to 10, their position in the calendar when it began around the spring equinox in March.[49] Others are uncertain. February may derive from the Februa festival or its eponymous februa ("purifications, expiatory offerings"), whose name may be either Sabine or preserve an archaic word for sulphuric.[53] April may relate to the Etruscan goddess Apru or the verb aperire ("to open").[citation needed] May and June may honor Maia[54] and Juno[55] or derive from archaic terms for "senior" and "junior". A few emperors attempted to add themselves to the calendar after Augustus, but without enduring success.
In classical Latin, the days of each month were usually reckoned as:[38]
| Day | Original 31-Day Months[h] | New Julian 31-Day Months[i] | New Julian 30-Day Months[j] | Original 29-Day Months[k] | February |
|---|---|---|---|---|---|
| 1 | Kal. | On the Kalends
Kalendis
| Kal. | Kal. | Kal. Feb. |
| 2 | a.d. VI Non. | The 4th Day before the Nones
ante diem quartum Nonas
| a.d. IV Non. | a.d. IV Non. | a.d. IV Non. Feb. |
| 3 | a.d. V Non. | The 3rd Day before the Nones
ante diem tertium Nonas
| a.d. III Non. | a.d. III Non. | a.d. III Non. Feb. |
| 4 | a.d. IV Non. | On the Day before the Nones
Pridie Nonas
| Prid. Non. | Prid. Non. | Prid. Non. Feb. |
| 5 | a.d. III Non. | On the Nones
Nonis
| Non. | Non. | Non. Feb. |
| 6 | Prid. Non. | The 8th Day before the Ides
ante diem octavum Idus
| a.d. VIII Eid. | a.d. VIII Eid. | a.d. VIII Eid. Feb. |
| 7 | Non. | The 7th Day before the Ides
ante diem septimum Idus
| a.d. VII Eid. | a.d. VII Eid. | a.d. VII Eid. Feb. |
| 8 | a.d. VIII Eid. | The 6th Day before the Ides
ante diem sextum Idus
| a.d. VI Eid. | a.d. VI Eid. | a.d. VI Eid. Feb. |
| 9 | a.d. VII Eid. | The 5th Day before the Ides
ante diem quintum Idus
| a.d. V Eid. | a.d. V Eid. | a.d. V Eid. Feb. |
| 10 | a.d. VI Eid. | The 4th Day before the Ides
ante diem quartum Idus
| a.d. IV Eid. | a.d. IV Eid. | a.d. IV Eid. Feb. |
| 11 | a.d. V Eid. | The 3rd Day before the Ides
ante diem tertium Idus
| a.d. III Eid. | a.d. III Eid. | a.d. III Eid. Feb. |
| 12 | a.d. IV Eid. | On the Day before the Ides
Pridie Idus
| Prid. Eid. | Prid. Eid. | Prid. Eid. Feb. |
| 13 | a.d. III Eid. | On the Ides
Idibus
| Eid. | Eid. | Eid. Feb. |
| 14 | Prid. Eid. | The 19th Day before the Kalends
ante diem undevicesimum Kalendas
| a.d. XVIII Kal. | a.d. XVII Kal. | a.d. XVI Kal. Mart. |
| 15 | Eid. | The 18th Day before the Kalends
ante diem duodevicesimum Kalendas
| a.d. XVII Kal. | a.d. XVI Kal. | a.d. XV Kal. Mart. |
| 16 | a.d. XVII Kal. | The 17th Day before the Kalends
ante diem septimum decimum Kalendas
| a.d. XVI Kal. | a.d. XV Kal. | a.d. XIV Kal. Mart. |
| 17 | a.d. XVI Kal. | The 16th Day before the Kalends
ante diem sextum decimum Kalendas
| a.d. XV Kal. | a.d. XIV Kal. | a.d. XIII Kal. Mart. |
| 18 | a.d. XV Kal. | The 15th Day before the Kalends
ante diem quintum decimum Kalendas
| a.d. XIV Kal. | a.d. XIII Kal. | a.d. XII Kal. Mart. |
| 19 | a.d. XIV Kal. | The 14th Day before the Kalends
ante diem quartum decimum Kalendas
| a.d. XIII Kal. | a.d. XII Kal. | a.d. XI Kal. Mart. |
| 20 | a.d. XIII Kal. | The 13th Day before the Kalends
ante diem tertium decimum Kalendas
| a.d. XII Kal. | a.d. XI Kal. | a.d. X Kal. Mart. |
| 21 | a.d. XII Kal. | The 12th Day before the Kalends
ante diem duodecimum Kalendas
| a.d. XI Kal. | a.d. X Kal. | a.d. IX Kal. Mart. |
| 22 | a.d. XI Kal. | The 11th Day before the Kalends
ante diem undecimum Kalendas
| a.d. X Kal. | a.d. IX Kal. | a.d. VIII Kal. Mart. |
| 23 | a.d. X Kal. | The 10th Day before the Kalends
ante diem decimum Kalendas
| a.d. IX Kal. | a.d. VIII Kal. | a.d. VII Kal. Mart. |
| 24 | a.d. IX Kal. | The 9th Day before the Kalends
ante diem nonum Kalendas
| a.d. VIII Kal. | a.d. VII Kal. | a.d. VI Kal. Mart.[l] |
| 25 | a.d. VIII Kal. | The 8th Day before the Kalends
ante diem octavum Kalendas
| a.d. VII Kal. | a.d. VI Kal. | a.d. V Kal. Mart. |
| 26 | a.d. VII Kal. | The 7th Day before the Kalends
ante diem septimum Kalendas
| a.d. VI Kal. | a.d. V Kal. | a.d. IV Kal. Mart. |
| 27 | a.d. VI Kal. | The 6th Day before the Kalends
ante diem sextum Kalendas
| a.d. V Kal. | a.d. IV Kal. | a.d. III Kal. Mart. |
| 28 | a.d. V Kal. | The 5th Day before the Kalends
ante diem quintum Kalendas
| a.d. IV Kal. | a.d. III Kal. | Prid. Kal. Mart. |
| 29 | a.d. IV Kal. | The 4th Day before the Kalends
ante diem quartum Kalendas
| a.d. III Kal. | Prid. Kal. | |
| 30 | a.d. III Kal. | The 3rd Day before the Kalends
ante diem tertium Kalendas
| Prid. Kal. | ||
| 31 | Prid. Kal. | On the Day Before the Kalends
Pridie Kalendas
|
Dates after the ides count forward to the kalends of the next month and are expressed as such. For example, March 19 was expressed as "the 14th day before the April Kalends" (a.d. XIV Kal. Apr.), without a mention of March itself. The day after a kalends, nones, or ides was also often expressed as the "day after" (postridie) owing to their special status as particularly unlucky "black days".
The anomalous status of the new 31-day months under the Julian calendar was an effect of Caesar's desire to avoid affecting the festivals tied to the nones and ides of various months. Because the dates at the ends of the month all counted forward to the next kalends, however, they were all shifted by one or two days by the change. This created confusion with regard to certain anniversaries. For instance, Augustus's birthday on the 23rd day of September was a.d. VIII Kal. Oct. in the old calendar but a.d. IX Kal. Oct.under the new system. The ambiguity caused honorary festivals to be held on either or both dates.
Intercalation[edit]
The Republican calendar only had 355 days, which meant that it would quickly unsynchronize from the solar year, causing, for example, agricultural festivals to occur out of season. The Roman solution to this problem was to periodically lengthen the calendar by adding extra days within February. February was broken into two parts, each with an odd number of days. The first part ended with the Terminalia on the 23rd (a.d. VII Kal. Mart.), which was considered the end of the religious year; the five remaining days beginning with the Regifugium on the 24th (a.d. VI Kal. Mart.) formed the second part; and the intercalary month Mercedonius was inserted between them. In such years, the days between the ides and the Regifugium were counted down to either the Intercalary Kalends or to the Terminalia. The intercalary month counted down to nones and ides on its 5th and 13th day in the manner of the other short months. The remaining days of the month counted down towards the March Kalends, so that the end of Mercedonius and the second part of February were indistinguishable to the Romans, one ending on a.d. VII Kal. Mart. and the other picking up at a.d. VI Kal. Mart. and bearing the normal festivals of such dates.
Apparently because of the confusion of these changes or uncertainty as to whether an intercalary month would be ordered, dates after the February ides are attested as sometimes counting down towards the Quirinalia (Feb. 17), the Feralia (Feb. 21), or Terminalia (Feb. 23)[56] rather than the intercalary or March kalends.
The third-century writer Censorinus says:
The fifth-century writer Macrobius says that the Romans intercalated 22 and 23 days in alternate years (Saturnalia, 1.13.12), the intercalation was placed after 23 February and the remaining five days of February followed (Saturnalia, 1.13.15). To avoid the nones falling on a nundine, where necessary an intercalary day was inserted "in the middle of the Terminalia, where they placed the intercalary month".[58]
This is historically correct. In 167 BC Intercalaris began on the day after 23 February [59] and in 170 BC it began on the second day after 23 February.[60] Varro, writing in the first century BC, says "the twelfth month was February, and when intercalations take place the five last days of this month are removed."[61] Since all the days after the Ides of Intercalaris were counted down to the beginning of March Intercalaris had either 27 days (making 377 for the year) or 28 (making 378 for the year).
There is another theory which says that in intercalary years February had 23 or 24 days and Intercalaris had 27. No date is offered for the Regifugium in 378-day years.[62]Macrobius describes a further refinement whereby, in one 8-year period within a 24-year cycle, there were only three intercalary years, each of 377 days. This refinement brings the calendar back in line with the seasons, and averages the length of the year to 365.25 days over 24 years.
The Pontifex Maximus determined when an intercalary month was to be inserted. On average, this happened in alternate years. The system of aligning the year through intercalary months broke down at least twice: the first time was during and after the Second Punic War. It led to the reform of the 191 bc Acilian Law on Intercalation, the details of which are unclear, but it appears to have successfully regulated intercalation for over a century. The second breakdown was in the middle of the first century bc and may have been related to the increasingly chaotic and adversarial nature of Roman politics at the time. The position of Pontifex Maximus was not a full-time job; it was held by a member of the Roman elite, who would almost invariably be involved in the machinations of Roman politics. Because the term of office of elected Roman magistrates was defined in terms of a Roman calendar year, a Pontifex Maximus would have reason to lengthen a year in which he or his allies were in power or shorten a year in which his political opponents held office.
Although there are many stories to interpret the intercalation, a period of 22 or 23 days is always ¾ synodic month. Obviously, the month beginning shifts forward (from the new moon, to the third quarter, to the full moon, to the first quarter, back the new moon) after intercalation.
Years[edit]
As mentioned above, Rome's legendary 10-month calendar notionally lasted for 304 days but was usually thought to make up the rest of the solar year during an unorganized winter period. The unattested but almost certain lunar year and the pre-Julian civil year were 354 or 355 days long, with the difference from the solar year more or less corrected by an irregular intercalary month. The Julian year was 365 days long, with a leap day doubled in length every fourth year, almost equivalent to the present Gregorian system.
The calendar era before and under the Roman kings is uncertain but dating by regnal years was common in antiquity. Under the Roman Republic, from 509 bc, years were most commonly described in terms of their reigning ordinary consuls.[27] (Temporary and honorary consuls were sometimes elected or appointed but were not used in dating.)[27] Consular lists were displayed on the public calendars. After the institution of the Roman Empire, regnal dates based on the emperors' terms in office became more common. Some historians of the later republic and early imperial eras dated from the legendary founding of the city of Rome (ab urbe condita or avc).[27] Varro's date for this was 753 bc but other writers used different dates, varying by several decades.[citation needed] Such dating was, however, never widespread. After the consuls waned in importance, most Roman dating was regnal[64] or followed Diocletian's 15-year Indiction tax cycle.[27] These cycles were not distinguished, however, so that "year 2 of the indiction" may refer to any of 298, 313, 328, &c.[27] The Orthodox subjects of the Byzantine Empire used various Christian eras, including those based on Diocletian's persecutions, Christ's incarnation, and the supposed age of the world.
The Romans did not have records of their early calendars but, like modern historians, assumed the year originally began in March on the basis of the names of the months following June. The consul M. Fulvius Nobilior (r. 189 bc) wrote a commentary on the calendar at the Temple of Hercules Musarum that claimed January had been named for Janus because the god faced both ways,[65][where?] suggesting it had been instituted as a first month.[citation needed] It was, however, usually said to have been instituted along with February, whose nature and festivals suggest it had originally been considered the last month of the year. The consuls' term of office—and thus the order of the years under the republic—seems to have changed several times. Their inaugurations were finally moved to 1 January (Kal. Ian.) in 153 bc to allow Q. Fulvius Nobilior to attack Segeda in Spain during the Celtiberian Wars, before which they had occurred on 15 March (Eid. Mart.).[66] There is reason to believe the inauguration date had been 1 May during the 3rd century bc until 222 bc[citation needed] and Livy mentions earlier inaugurations on 15 May (Eid. Mai.), 1 July (Kal. Qui.), 1 August (Kal. Sex.), 1 October (Kal. Oct.), and 15 December (Eid. Dec.).[67][where?] Under the Julian calendar, the year began on 1 January but years of the Indiction cycle began on 1 September.
In addition Egypt's separate calendar, some provinces maintained their records using a local era.[27] Africa dated its records sequentially from 39 bc;[64] Spain from ad 38.[citation needed] This dating system continued as the Spanish era used in medieval Spain.[citation needed]
Conversion to Julian or Gregorian dates[edit]
The continuity of names from the Roman to the Gregorian calendar can lead to the mistaken belief that Roman dates correspond to Julian or Gregorian ones. In fact, the essentially complete list of Roman consuls allows general certainty of years back to the establishment of the republic but the uncertainty as to the end of lunar dating and the irregularity of Roman intercalation means that dates which can be independently verified are invariably weeks to months outside of their "proper" place. Two astronomical events dated by Livy show the calendar 4 months out of alignment with the Julian date in 190 bc and 2 months out of alignment in 168 bc. Thus, "the year of the consulship of Publius Cornelius Sciopio Africanus and Publius Licinius Crassus" (usually given as "205 bc") actually began on 15 March 205 bc and ended on 14 March 204 bc according to the Roman calendar but may have begun as early as November or December 206 bc owing to its misalignment. Even following the establishment of the Julian calendar, the leap years were not applied correctly by the Roman priests, meaning dates are a few days out of their "proper" place until a few decades into Augustus's reign.
Given the paucity of records regarding the state of the calendar and its intercalation, historians have reconstructed the correspondence of Roman dates to their Julian and Gregorian equivalents from disparate sources. There are detailed accounts of the decades leading up to the Julian reform, particularly the speeches and letters of Cicero, which permit an established chronology back to about 58 bc. The nundinal cycle and a few known synchronisms—e.g., a Roman date in terms of the Attic calendar and Olympiad—are used to generate contested chronologies back to the start of the First Punic War in 264 bc. Beyond that, dates are roughly known based on clues such as the dates of harvestsand seasonal religious festivals.
Ides of March
The Ides of March (Latin: Idus Martiae, Late Latin: Idus Martii)[1] is a day on the Roman calendar that corresponds to March 15. It was marked by several religious observances and was notable for the Romans as a deadline for settling debts[2]. In 44 BC, it became notorious as the date of the assassination of Julius Caesar. The death of Caesar made the Ides of March a turning point in Roman history, as one of the events that marked the transition from the historical period known as the Roman Republic to the Roman Empire.[3]
Although March (Martius) was the third month of the Julian calendar, in the oldest Roman calendar it was the first month of the year. The holidays observed by the Romans from the first through the Ides often reflect their origin as new-year celebrations.[citation needed]
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[hide]Ides[edit]
The Romans did not number days of a month sequentially from the first through the last day. Instead, they counted back from three fixed points of the month: the Nones (5th or 7th, depending on the length of the month), the Ides (13th or 15th), and the Kalends (1st of the following month). The Ides occurred near the midpoint, on the 13th for most months, but on the 15th for March, May, July, and October. The Ides were supposed to be determined by the full moon, reflecting the lunar origin of the Roman calendar. On the earliest calendar, the Ides of March would have been the first full moon of the new year.[4]
Religious observances[edit]
The Ides of each month were sacred to Jupiter, the Romans' supreme deity. The Flamen Dialis, Jupiter's high priest, led the "Ides sheep" (ovis Idulius) in procession along the Via Sacra to the arx, where it was sacrificed.[5]
In addition to the monthly sacrifice, the Ides of March was also the occasion of the Feast of Anna Perenna, a goddess of the year (Latin annus) whose festival originally concluded the ceremonies of the new year. The day was enthusiastically celebrated among the common people with picnics, drinking, and revelry.[6] One source from late antiquity also places the Mamuralia on the Ides of March.[7] This observance, which has aspects of scapegoat or ancient Greek pharmakos ritual, involved beating an old man dressed in animal skins and perhaps driving him from the city. The ritual may have been a new year festival representing the expulsion of the old year.[8][9]
In the later Imperial period, the Ides began a "holy week" of festivals[10][11][12] for Cybele and Attis. The Ides was the day of Canna intrat("The Reed enters"), when Attis was born and exposed as an infant among the reeds of a Phrygian river.[13] He was discovered—depending on the version of the myth—by either shepherds or the goddess Cybele, who was also known as the Magna Mater ("Great Mother").[14] A week later, on 22 March, the day of Arbor intrat ("The Tree enters") commemorated the death of Attis under a pine tree. A college of priests called dendrophoroi ("tree bearers") cut down a tree,[15] suspended from it an image of Attis,[16] and carried it to the temple of the Magna Mater with lamentations. The day was formalized as part of the official Roman calendar under Claudius.[17] A three-day period of mourning followed,[18] culminating with the rebirth of Attis on 25 March, the date of the vernal equinox on the Julian calendar.[19]
Assassination of Caesar[edit]
In modern times, the Ides of March is best known as the date on which Julius Caesar was assassinated in 44 BC. Caesar was stabbed to death at a meeting of the Senate. As many as 60 conspirators, led by Brutus and Cassius, were involved. According to Plutarch,[20] a seer had warned that harm would come to Caesar no later than the Ides of March. On his way to the Theatre of Pompey, where he would be assassinated, Caesar passed the seer and joked, "The Ides of March are come", implying that the prophecy had not been fulfilled, to which the seer replied "Aye, Caesar; but not gone."[20] This meeting is famously dramatised in William Shakespeare's play Julius Caesar, when Caesar is warned by the soothsayer to "beware the Ides of March."[21][22] The Roman biographer Suetonius[23]identifies the "seer" as a haruspex named Spurinna.
Caesar's death was a closing event in the crisis of the Roman Republic, and triggered the civil war that would result in the rise to sole power of his adopted heir Octavian (later known as Augustus).[24] Writing under Augustus, Ovid portrays the murder as a sacrilege, since Caesar was also the Pontifex Maximus of Rome and a priest of Vesta.[25] On the fourth anniversary of Caesar's death in 40 BC, after achieving a victory at the siege of Perugia, Octavian executed 300 senators and knights who had fought against him under Lucius Antonius, the brother of Mark Antony.[26] The executions were one of a series of actions taken by Octavian to avenge Caesar's death. Suetonius and the historian Cassius Dio characterised the slaughter as a religious sacrifice,[27][28] noting that it occurred on the Ides of March at the new altar to the deified Julius.
USS Kitty Hawk riot
USS Kitty Hawk c. 1975
| |
| Date | 12/13 October 1972[1] |
|---|---|
| Location | off Vietnam |
| Participants | Kitty Hawk crew |
| Outcome | 45–60 crewmembers injured (3 seriously) |
The USS Kitty Hawk riot was part of widespread antiwar protests within the U.S. military which took place as part of a movement called SOS (Stop Our Ships/Support Our Sailors) on the United States Navy aircraft carrier Kitty Hawk on the night of 12/13 October 1972, off the coast of Vietnam while participating in Operation Linebacker of the Vietnam War. A report by the House Armed Services Committee concluded that the rebellion had been precipitated by orders received to return to Vietnam from Subic Bay. These orders were given after incidents of sabotage by U.S. sailors had disabled USS Ranger and USS Forrestal. Kitty Hawkwas eventually forced to retire to San Diego and was removed from the war.[2]
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[hide]Background[edit]
The riot was led by African-American crew members who responded violently when Marines attempted to disrupt their protest meetings. Three had to be evacuated to shore hospitals for further treatment. Forty-five to sixty Kitty Hawk crewmen were injured in total. The ship's captain, Marland Townsend, and executive officer, Commander Benjamin Cloud (who was African-American), dissuaded the rioters from further violence. This allowed the carrier to launch her Linebacker air missions as scheduled on the morning of 12 October. Nineteen of the rioters were later found guilty by the Navy of at least one charge connected to the riot.
Aftermath[edit]
The incident was publicized in The New York Times. Subsequent unrest on Kitty Hawk′s sister ship Constellation was, in a similar fashion, described by the media as a "racial outbreak" in an effort to downplay the antiwar movement within the armed forces. However, many of its crew were active participants in the SOS movement, having earlier produced a petition with 1,500 signatures to allow Jane Fonda's antiwar show to perform on board, produced their own antiwar newsletter (as did Kitty Hawk, entitled Kitty Litter), and supported dozens of servicemen who refused to board for Vietnam duty.
Nim
Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap. The goal of the game is to avoid being the player who must remove the last object.
Variants of Nim have been played since ancient times.[1] The game is said to have originated in China—it closely resembles the Chinese game of 捡石子 jiǎn-shízi, or "picking stones"[2]—but the origin is uncertain; the earliest European references to Nim are from the beginning of the 16th century. Its current name was coined by Charles L. Bouton of Harvard University, who also developed the complete theory of the game in 1901,[3] but the origins of the name were never fully explained.
Nim is typically played as a misère game, in which the player to take the last object loses. Nim can also be played as a normal playgame, where the player taking the last object wins. This is called normal play because the last move is a winning move in most games; Nim is usually played so that the last move loses.
Normal play Nim (or more precisely the system of nimbers) is fundamental to the Sprague–Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a Nim heap that yields the same outcome when played in parallel with other normal play impartial games (see disjunctive sum).
While all normal play impartial games can be assigned a Nim value, that is not the case under the misère convention. Only tame games can be played using the same strategy as misère nim.
At the 1940 New York World's Fair Westinghouse displayed a machine, the Nimatron, that played Nim.[4] From May 11, 1940 to October 27, 1940 only a few people were able to beat the machine in that six week period, if they did they were presented with a coin that said Nim Champ.[5][6] It was also one of the first ever electronic computerized games. Ferranti built a Nim playing computer which was displayed at the Festival of Britain in 1951. In 1952 Herbert Koppel, Eugene Grant and Howard Bailer, engineers from the W. L. Maxon Corporation, developed a machine weighing 50 pounds which played Nim against a human opponent and regularly won.[7] A Nim Playing Machine has been described made from TinkerToy.[8]
The game of Nim was the subject of Martin Gardner's February 1958 Mathematical Games column in Scientific American. A version of Nim is played—and has symbolic importance—in the French New Wave film Last Year at Marienbad (1961).[9]
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[hide]Game play and illustration[edit]
The normal game is between two players and played with three heaps of any number of objects. The two players alternate taking any number of objects from any single one of the heaps. The goal is to be the last to take an object. In misère play, the goal is instead to ensure that the opponent is forced to take the last remaining object.
The following example game is played between fictional players Bob and Alice who start with heaps of three, four and five objects.
Sizes of heaps Moves A B C 3 4 5 Bob takes 2 from A 1 4 5 Alice takes 3 from C 1 4 2 Bob takes 1 from B 1 3 2 Alice takes 1 from B 1 2 2 Bob takes entire A heap, leaving two 2s. 0 2 2 Alice takes 1 from B 0 1 2 Bob takes 1 from C leaving two 1s. (In misère play he would take 2 from C leaving (0, 1, 0).) 0 1 1 Alice takes 1 from B 0 0 1 Bob takes entire C heap and wins.
Winning positions[edit]
The practical strategy to win at the game of Nim is for a player to get the other into one of the following positions, and every successive turn afterwards they should be able to make one of the lower positions. Only the last move changes between misere and normal play.
| 2 Heaps | 3 Heaps | 4 Heaps |
|---|---|---|
| 1 1 * | 1 1 1 ** | 1 1 1 1 * |
| 2 2 | 1 2 3 | 1 1 n n |
| 3 3 | 1 4 5 | 1 2 4 7 |
| 4 4 | 1 6 7 | 1 2 5 6 |
| 5 5 | 1 8 9 | 1 3 4 6 |
| 6 6 | 2 4 6 | 1 3 5 7 |
| 7 7 | 2 5 7 | 2 3 4 5 |
| 8 8 | 3 4 7 | 2 3 6 7 |
| 9 9 | 3 5 6 | 2 3 8 9 |
| (n n) | 4 8 12 | 4 5 6 7 |
| 4 9 13 | 4 5 8 9 | |
| 5 8 13 | n n m m | |
| 5 9 12 | n n n n |
* Only valid for normal play.
** Only valid for misere. For the generalisations, n and m can be any value > 0, and they may be the same.
Mathematical theory[edit]
Nim has been mathematically solved for any number of initial heaps and objects, and there is an easily calculated way to determine which player will win and what winning moves are open to that player.
The key to the theory of the game is the binary digital sum of the heap sizes, that is, the sum (in binary) neglecting all carries from one digit to another. This operation is also known as "exclusive or" (xor) or "vector addition over GF(2)". Within combinatorial game theory it is usually called the nim-sum, as it will be called here. The nim-sum of x and yis written x ⊕ y to distinguish it from the ordinary sum, x + y. An example of the calculation with heaps of size 3, 4, and 5 is as follows:
Binary Decimal 0112 310 Heap A 1002 410 Heap B 1012 510 Heap C --- 0102 210 The nim-sum of heaps A, B, and C, 3 ⊕ 4 ⊕ 5 = 2
An equivalent procedure, which is often easier to perform mentally, is to express the heap sizes as sums of distinct powers of 2, cancel pairs of equal powers, and then add what's left:
3 = 0 + 2 + 1 = 2 1 Heap A 4 = 4 + 0 + 0 = 4 Heap B 5 = 4 + 0 + 1 = 4 1 Heap C -------------------------------------------------------------------- 2 = 2 What's left after canceling 1s and 4s
In normal play, the winning strategy is to finish every move with a nim-sum of 0. This is always possible if the nim-sum is not zero before the move. If the nim-sum is zero, then the next player will lose if the other player does not make a mistake. To find out which move to make, let X be the nim-sum of all the heap sizes. Find a heap where the nim-sum of X and heap-size is less than the heap-size - the winning strategy is to play in such a heap, reducing that heap to the nim-sum of its original size with X. In the example above, taking the nim-sum of the sizes is X = 3 ⊕ 4 ⊕ 5 = 2. The nim-sums of the heap sizes A=3, B=4, and C=5 with X=2 are
- A ⊕ X = 3 ⊕ 2 = 1 [Since (011) ⊕ (010) = 001 ]
- B ⊕ X = 4 ⊕ 2 = 6
- C ⊕ X = 5 ⊕ 2 = 7
The only heap that is reduced is heap A, so the winning move is to reduce the size of heap A to 1 (by removing two objects).
As a particular simple case, if there are only two heaps left, the strategy is to reduce the number of objects in the bigger heap to make the heaps equal. After that, no matter what move your opponent makes, you can make the same move on the other heap, guaranteeing that you take the last object.
When played as a misère game, Nim strategy is different only when the normal play move would leave no heap of size two or larger. In that case, the correct move is to leave an odd number of heaps of size one (in normal play, the correct move would be to leave an even number of such heaps).
In a misère game with heaps of sizes three, four and five, the strategy would be applied like this:
A B C nim-sum
3 4 5 0102=210 I take 2 from A, leaving a sum of 000, so I will win.
1 4 5 0002=010 You take 2 from C
1 4 3 1102=610 I take 2 from B
1 2 3 0002=010 You take 1 from C
1 2 2 0012=110 I take 1 from A
0 2 2 0002=010 You take 1 from C
0 2 1 0112=310 The normal play strategy would be to take 1 from B, leaving an even number (2)
heaps of size 1. For misère play, I take the entire B heap, to leave an odd
number (1) of heaps of size 1.
0 0 1 0012=110 You take 1 from C, and lose.
Example implementation[edit]
The previous strategy for a misère game can be easily implemented (for example in Python, below).
import functools
MISERE = 'misere'
NORMAL = 'normal'
def nim(heaps, game_type):
"""
Computes next move for Nim, for both game types normal and misere.
if there is a winning move:
return tuple(heap_index, amount_to_remove)
else:
return "You will lose :("
- mid-game scenarios are the same for both game types
>>> print(nim([1, 2, 3], MISERE))
misere [1, 2, 3] You will lose :(
>>> print(nim([1, 2, 3], NORMAL))
normal [1, 2, 3] You will lose :(
>>> print(nim([1, 2, 4], MISERE))
misere [1, 2, 4] (2, 1)
>>> print(nim([1, 2, 4], NORMAL))
normal [1, 2, 4] (2, 1)
- endgame scenarios change depending upon game type
>>> print(nim([1], MISERE))
misere [1] You will lose :(
>>> print(nim([1], NORMAL))
normal [1] (0, 1)
>>> print(nim([1, 1], MISERE))
misere [1, 1] (0, 1)
>>> print(nim([1, 1], NORMAL))
normal [1, 1] You will lose :(
>>> print(nim([1, 5], MISERE))
misere [1, 5] (1, 5)
>>> print(nim([1, 5], NORMAL))
normal [1, 5] (1, 4)
"""
print(game_type, heaps, end=' ')
is_misere = game_type == MISERE
is_near_endgame = False
count_non_0_1 = sum(1 for x in heaps if x > 1)
is_near_endgame = (count_non_0_1 <= 1)
# nim sum will give the correct end-game move for normal play but
# misere requires the last move be forced onto the opponent
if is_misere and is_near_endgame:
moves_left = sum(1 for x in heaps if x > 0)
is_odd = (moves_left % 2 == 1)
sizeof_max = max(heaps)
index_of_max = heaps.index(sizeof_max)
if sizeof_max == 1 and is_odd:
return "You will lose :("
# reduce the game to an odd number of 1's
return index_of_max, sizeof_max - int(is_odd)
nim_sum = functools.reduce(lambda x, y: x ^ y, heaps)
if nim_sum == 0:
return "You will lose :("
# Calc which move to make
for index, heap in enumerate(heaps):
target_size = heap ^ nim_sum
if target_size < heap:
amount_to_remove = heap - target_size
return index, amount_to_remove
if __name__ == "__main__":
import doctest
doctest.testmod()
Proof of the winning formula[edit]
The soundness of the optimal strategy described above was demonstrated by C. Bouton.
Theorem. In a normal Nim game, the player making the first move has a winning strategy if and only if the nim-sum of the sizes of the heaps is nonzero. Otherwise, the second player has a winning strategy.
Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+) and also satisfies an additional property, x ⊕ x = 0 (technically speaking, that the nonnegative integers under ⊕ form an Abelian group of exponent 2).
Let x1, ..., xn be the sizes of the heaps before a move, and y1, ..., yn the corresponding sizes after a move. Let s = x1 ⊕ ... ⊕ xn and t = y1 ⊕ ... ⊕ yn. If the move was in heap k, we have xi = yi for all i ≠ k, and xk > yk. By the properties of ⊕ mentioned above, we have
t = 0 ⊕ t
= s ⊕ s ⊕ t
= s ⊕ (x1 ⊕ ... ⊕ xn) ⊕ (y1 ⊕ ... ⊕ yn)
= s ⊕ (x1 ⊕ y1) ⊕ ... ⊕ (xn ⊕ yn)
= s ⊕ 0 ⊕ ... ⊕ 0 ⊕ (xk ⊕ yk) ⊕ 0 ⊕ ... ⊕ 0
= s ⊕ xk ⊕ yk
(*) t = s ⊕ xk ⊕ yk.
The theorem follows by induction on the length of the game from these two lemmas.
Lemma 1. If s = 0, then t ≠ 0 no matter what move is made.
Proof: If there is no possible move, then the lemma is vacuously true (and the first player loses the normal play game by definition). Otherwise, any move in heap k will produce t = xk ⊕ yk from (*). This number is nonzero, since xk ≠ yk.
Lemma 2. If s ≠ 0, it is possible to make a move so that t = 0.
Proof: Let d be the position of the leftmost (most significant) nonzero bit in the binary representation of s, and choose k such that the dth bit of xk is also nonzero. (Such a k must exist, since otherwise the dth bit of s would be 0.) Then letting yk = s ⊕ xk, we claim that yk < xk: all bits to the left of d are the same in xk and yk, bit d decreases from 1 to 0 (decreasing the value by 2d), and any change in the remaining bits will amount to at most 2d−1. The first player can thus make a move by taking xk − yk objects from heap k, then
t = s ⊕ xk ⊕ yk (by (*)) = s ⊕ xk ⊕ (s ⊕ xk) = 0.
The modification for misère play is demonstrated by noting that the modification first arises in a position that has only one heap of size 2 or more. Notice that in such a position s≠ 0, therefore this situation has to arise when it is the turn of the player following the winning strategy. The normal play strategy is for the player to reduce this to size 0 or 1, leaving an even number of heaps with size 1, and the misère strategy is to do the opposite. From that point on, all moves are forced.
Variations[edit]
Dividing natural number[edit]
Give any natural number n, the two people can divide n by a prime power (
A000961) which is a factor of n (except 1), the person who gets 1 wins (or loses).
A000961) which is a factor of n (except 1), the person who gets 1 wins (or loses).
If , where is the k-th prime, then it is a Nim game with k groups of stones, and the r-th groups has stones.
If the divisor changes to "a power of squarefree numbers" ( A072774) except 1, it is Wythoff's game.
A072774) except 1, it is Wythoff's game.
The divisor can also change to "a divisor of m" for fixed m, where m is a divisor of n. (m should be divisible by all of the prime factors of n and should be less than n)
Of course, you can choose a set of allowed divisors. For example, {2, 3, 4, 12, 15, 20, 24, 25, 30, 32, 36}.
The subtraction game S(1, 2, . . ., k)[edit]
In another game which is commonly known as Nim (but is better called the subtraction game S (1,2,...,k)), an upper bound is imposed on the number of objects that can be removed in a turn. Instead of removing arbitrarily many objects, a player can only remove 1 or 2 or ... or k at a time. This game is commonly played in practice with only one heap (for instance with k = 3 in the game Thai 21 on Survivor: Thailand, where it appeared as an Immunity Challenge).
Bouton's analysis carries over easily to the general multiple-heap version of this game. The only difference is that as a first step, before computing the Nim-sums, we must reduce the sizes of the heaps modulo k + 1. If this makes all the heaps of size zero (in misère play), the winning move is to take k objects from one of the heaps. In particular, in ideal play from a single heap of n objects, the second player can win if and only if
- n ≡ 0 (mod k + 1) (in normal play), or
- n ≡ 1 (mod k + 1) (in misère play).
This follows from calculating the nim-sequence of S(1,2,...,k),
from which the strategy above follows by the Sprague–Grundy theorem.
The 21 game[edit]
The game "21" is played as a misère game with any number of players who take turns saying a number. The first player says "1" and each player in turn increases the number by 1, 2, or 3, but may not exceed 21; the player forced to say "21" loses. This can be modeled as a subtraction game with a heap of 21–n objects. The winning strategy for the two-player version of this game is to always say a multiple of 4; it is then guaranteed that the other player will ultimately have to say 21 – so in the standard version where the first player opens with "1", they start with a losing move.
The 21 game can also be played with different numbers, like "Add at most 5; lose on 34".
A sample game of 21 in which the second player follows the winning strategy:
Player Number 1 1 2 4 1 5, 6 or 7 2 8 1 9, 10 or 11 2 12 1 13, 14 or 15 2 16 1 17, 18 or 19 2 20 1 21
The 100 game[edit]
A similar version is the "100 game": two players start from 0 and alternatively add a number from 1 to 10 to the sum. The player who reaches 100 wins. The winning strategy is to reach a number in which the digits are subsequent (e.g. 01, 12, 23, 34,...) and control the game by jumping through all the numbers of this sequence. Once reached 89, the opponent has lost (he can only tell numbers from 90 to 99, and the next answer can in any case be 100).
A multiple-heap rule[edit]
In another variation of Nim, besides removing any number of objects from a single heap, one is permitted to remove the same number of objects from each heap.
Circular Nim[edit]
Yet another variation of Nim is 'Circular Nim', where any number of objects are placed in a circle, and two players alternately remove one, two or three adjacent objects. For example, starting with a circle of ten objects,
. . . . . . . . . .
three objects are taken in the first move
_ . . . . . . . _ _
then another three
_ . _ _ _ . . . _ _
then one
_ . _ _ _ . . _ _ _
but then three objects cannot be taken out in one move.
Grundy's game[edit]
In Grundy's game, another variation of Nim, a number of objects are placed in an initial heap, and two players alternately divide a heap into two nonempty heaps of different sizes. Thus, six objects may be divided into piles of 5+1 or 4+2, but not 3+3. Grundy's game can be played as either misère or normal play.
Greedy Nim[edit]
Greedy Nim is a variation where the players are restricted to choosing stones from only the largest pile.[10] It is a finite impartial game. Greedy Nim Misère has the same rules as Greedy Nim, but here the last player able to make a move loses.
Let the largest number of stones in a pile be m, the second largest number of stones in a pile be n. Let pm be the number of piles having m stones, pn be the number of piles having n stones. Then there is a theorem that game positions with pm even are P positions. [11] This theorem can be shown by considering the positions where pm is odd. If pm is larger than 1, all stones may be removed from this pile to reduce pm by 1 and the new pm will be even. If pm = 1 (i.e. the largest heap is unique), there are two cases:
- If pn is odd, the size of the largest heap is reduced to n (so now the new pm is even).
- If pn is even, the largest heap is removed entirely, leaving an even number of largest heaps.
Thus there exists a move to a state where pm is even. Conversely, if pm is even, if any move is possible (pm ≠ 0) then it must take the game to a state where pm is odd. The final position of the game is even (pm = 0). Hence each position of the game with pm even must be a P position.
Index-k Nim[edit]
A generalization of multi-heap Nim was called "Nim" or "index-k" Nim by E. H. Moore,[12] who analyzed it in 1910. In index-k Nim, instead of removing objects from only one heap, players can remove objects from at least one but up to k different heaps. The number of elements that may be removed from each heap may be either arbitrary, or limited to at most r elements, like in the "subtraction game" above.
The winning strategy is as follows: Like in ordinary multi-heap Nim, one considers the binary representation of the heap sizes (or heap sizes modulo r + 1). In ordinary Nim one forms the XOR-sum (or sum modulo 2) of each binary digit, and the winning strategy is to make each XOR sum zero. In the generalization to index-k Nim, one forms the sum of each binary digit modulo k + 1.
Again the winning strategy is to move such that this sum is zero for every digit. Indeed, the value thus computed is zero for the final position, and given a configuration of heaps for which this value is zero, any change of at most k heaps will make the value non-zero. Conversely, given a configuration with non-zero value, one can always take from at most k heaps, carefully chosen, so that the value will become zero.
Building Nim[edit]
Building Nim is a variant of Nim where the two players first construct the game of Nim. Given n stones and s empty piles, the players alternate turns placing exactly one stone into a pile of their choice.[13] Once all the stones are placed, a game of Nim begins, starting with the next player that would move. This game is denoted BN(n,s).
Jupiter Calling (album)
| Jupiter Calling | ||||
|---|---|---|---|---|
 | ||||
| Studio album by The Corrs | ||||
| Released | 10 November 2017 | |||
| Recorded | RAK Studios[1] | |||
| Length | 55:17 | |||
| Label | East West | |||
| Producer | T Bone Burnett[2] | |||
| The Corrs chronology | ||||
| ||||
| Singles from Jupiter Calling | ||||
| ||||
Jupiter Calling is the seventh album by The Corrs, released on 10 November 2017 by East West Records. It is their first new material in two years, following White Light (2015).[2] It was produced by T Bone Burnett,[2] and recorded at RAK Studios in London.[1] It was promoted by a show at the Royal Albert Hall on 19 October,[3] for which tickets were made available on 15 September.[4]
Contents
[hide]Background[edit]
Following the release of White Light in 2015, the band toured the UK and Ireland in January 2016, and continental Europe in mid-2016 on their White Light Tour. Andrea Corr spoke to The Sun in January 2016, stating that the band were not planning to record a follow-up soon after, "as long as we're all happy to continue. And we are all very happy."[5]
Recording and content[edit]
The band chose to include a song about Syria on the album, called "SOS (Song of Syria)", which they called their most "politically outspoken and evocative" song to date.[6] Andrea also spoke of playing the song "Son of Solomon" for Burnett: "He said 'Okay, don't play that any more'. When you're on the verge of knowing something you're much better than when you know it too well. And he was right. That's when the magic happens."[6] Caroline Corr described the recording process as "the most freeing experience we've had in the studio."[1]
Of working with the band, Burnett complimented their "deep, generous spirit" and their "writing, singing and playing".[6] The album was primarily recorded live, with minimal overdubbing.[1] While working on the album, Burnett made use of 40 spools of two-inch tape and a Ludwig drum kit.[1] Guitarist Anthony Drennan and bassist Robbie Malone are set to appear on the record.[1]
Singles[edit]
On 21 September 2017, "Son of Solomon" was released as the first promotional single from the album. The song was accompanied by a homemade video, recorded and edited by Andrea Corr. "SOS (Song of Syria)" was released as the lead single on 29 September 2017. The Corrs premiered the song on BBC Radio 2 on 28 September 2017[7] and it was immediately released to YouTube after the radio premiere.
Critical reception[edit]
| Professional ratings | |
|---|---|
| Aggregate scores | |
| Source | Rating |
| Metacritic | 58/100[8] |
| Review scores | |
| Source | Rating |
| AllMusic |    [9] |
| The Guardian | [10] |
| The Irish Times | [11] |
Jupiter Calling received generally mixed reviews from music critics. At Metacritic, which assigns a normalized rating out of 100 to reviews from mainstream publications, the album received an average score of 58 based on 4 reviews, indicating "mixed or average reviews".[8]
Track listing[edit]
Track listing adapted from iTunes.[12]
All tracks written by The Corrs.
| No. | Title | Length |
|---|---|---|
| 1. | "Son of Solomon" | 4:21 |
| 2. | "Chasing Shadows" | 3:36 |
| 3. | "Bulletproof Love" | 3:17 |
| 4. | "Road to Eden" | 4:41 |
| 5. | "Butter Flutter" | 3:44 |
| 6. | "SOS" | 3:36 |
| 7. | "Dear Life" | 4:18 |
| 8. | "No Go Baby" | 2:46 |
| 9. | "Hit My Ground Running" | 4:39 |
| 10. | "Live Before I Die" | 4:10 |
| 11. | "Season of Our Love" | 3:43 |
| 12. | "A Love Divine" | 4:35 |
| 13. | "The Sun and the Moon" | 7:51 |
Charts[edit]
| Chart (2017) | Peak position |
|---|---|
| Australian Albums (ARIA)[13] | 27 |
| Austrian Albums (Ö3 Austria)[14] | 46 |
| Belgian Albums (Ultratop Flanders)[15] | 30 |
| Belgian Albums (Ultratop Wallonia)[16] | 43 |
| Czech Albums (ČNS IFPI)[17] | 39 |
| Dutch Albums (MegaCharts)[18] | 31 |
| French Albums (SNEP)[19] | 52 |
| German Albums (Offizielle Top 100)[20] | 42 |
| Irish Albums (IRMA)[21] | 20 |
| New Zealand Heatseeker Albums (RMNZ)[22] | 4 |
| Portuguese Albums (AFP)[23] | 49 |
| Scottish Albums (OCC)[24] | 14 |
| Spanish Albums (PROMUSICAE)[25] | 13 |
| Swiss Albums (Schweizer Hitparade)[26] | 25 |
| UK Albums (OCC)[27] | 15 |
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