by neufer » Fri Sep 11, 2020 4:34 pm
bystander wrote: ↑Fri Sep 11, 2020 2:41 pm
Ann wrote: ↑Fri Sep 11, 2020 2:18 pm
orin stepanek wrote: ↑Fri Sep 11, 2020 11:32 am
Aw! Mars was just playing hide & seek! It came out at
ali ali free!
https://en.wikipedia.org/wiki/Archimedes%27s_cattle_problem wrote:
<<Archimedes's cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions. The problem was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of forty-four lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773.
The problem, from an abridgement of the German translations published by Georg Nesselmann in 1842, and by Krumbiegel in 1880, states:
...................................................................................
Compute, O friend, the number of the cattle of the sun which once grazed upon the plains of Sicily, divided according to color into four herds, one milk-white, one black, one dappled and one yellow. The number of bulls is greater than the number of cows, and the relations between them are as follows:
White bulls = ( 1/2 + 1/3 ) black bulls + yellow bulls,
Black bulls = ( 1/4 + 1/5 ) dappled bulls + yellow bulls,
Dappled bulls = ( 1/6 + 1/7 ) white bulls + yellow bulls,
White cows = ( 1/3 + 1/4 ) black herd,
Black cows = ( 1/4 + 1/5 ) dappled herd,
Dappled cows = ( 1/5 + 1/6 ) yellow herd,
Yellow cows = ( 1/6 + 1/7 ) white herd.
If thou canst give, O friend, the number of each kind of bulls and cows, thou art no novice in numbers, yet can not be regarded as of high skill. Consider, however, the following additional relations between the bulls of the sun:
White bulls + black bulls = a square number,
Dappled bulls + yellow bulls = a triangular number.
If thou hast computed these also, O friend, and found the total number of cattle,
then exult as a conqueror, for thou hast proved thyself most skilled in numbers.
...................................................................................
The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by Carl Ernst August Amthor (1845–1916), headmaster of the Gymnasium zum Heiligen Kreuz (Gymnasium of the Holy Cross) in Dresden, Germany. Using logarithmic tables, he calculated the first digits of the smallest solution,
showing that it is about 7.76 × 10206544 cattle,
far more than could fit in the observable universe.>>
https://en.wikipedia.org/wiki/Cattle_of_Helios wrote:
<<In Greek mythology, the Cattle of Helios (Greek: Ἠελίοιο βόες, Hēelíoio bóes), also called the
Oxen of the Sun, are cattle pastured on the island of Thrinacia (sometimes identified with Sicily).
Helios, also known as the titan of the sun, is said to have had seven herds of oxen and seven flocks of sheep, each numbering fifty head. In the Odyssey, Homer describes these immortal cattle as handsome (ἄριστος), wide-browed, fat (εúρυμέτωπος) and straight-horned (ὀρθόκραιρος). The cattle were guarded by Helios’ daughters, Phaëthusa and Lampetië, and it was known by all that any harm to any single animal was sure to bring down the wrath of the titan.
Tiresias and Circe both warn Odysseus to shun the isle of Helios. When Eurylochus begs to be allowed to land to prepare supper, Odysseus grudgingly agrees on condition that the crew swear that if they come upon a herd of cattle or a great flock of sheep, no one will kill any of them. They are held on the isle for a month by an unfavorable storm sent by Poseidon.
When Odysseus goes up the island to pray to the gods and ask for help, Eurylochus convinces the crew to drive off the best of the cattle of Helios and sacrifice them to the gods: "if he be somewhat wroth for his cattle with straight horns, and is fain to wreck our ship, and the other gods follow his desire, rather with one gulp at the wave would I cast my life away, than be slowly straitened to death in a desert isle." When he returns to the ship, Odysseus rebukes his companions for disobeying his orders. But it is too late, the cattle are dead and gone.
Lampetie tells Helios that Odysseus' men have slain his cattle. In turn, Helios orders the gods to take vengeance on Odysseus' men. He threatens that if they do not pay him full atonement for the cattle, he will take the sun to the Underworld and shine it among the dead. Zeus promises Helios to smite their ship with a lightning bolt and cleave it in pieces in the midst of the ocean.
Soon the gods show signs and wonders to the Odysseus' men. The skins begin creeping and the flesh bellowing upon the spits, both the roast and raw, and there is a sound like the voice of cattle. For six days, Odysseus's company feast on the kine of Helios. On the seventh day, the wind changes. After they set sail, Zeus keeps his word and the ship is destroyed by lightning during a storm and all of his men die. Odysseus escapes by swimming to Calypso's island.>>
[quote=bystander post_id=306128 time=1599835272 user_id=112005]
[quote=Ann post_id=306127 time=1599833931 user_id=129702]
[quote="orin stepanek" post_id=306126 time=1599823965 user_id=100812]
Aw! Mars was just playing hide & seek! It came out at [b][color=#FF8000]ali ali free[/color][/b]! :lol2:[/quote]
[list]🤷♂️¯\_(ツ)_/¯ :? :?: [/list][/quote]
[list][url=https://en.wikipedia.org/wiki/Olly_olly_oxen_free]Olly olly oxen free[/url][/list][/quote][quote=https://en.wikipedia.org/wiki/Archimedes%27s_cattle_problem]
<<Archimedes's cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions. The problem was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of forty-four lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773.
The problem, from an abridgement of the German translations published by Georg Nesselmann in 1842, and by Krumbiegel in 1880, states:
...................................................................................
[b][i][color=#0000FF] Compute, O friend, the number of the cattle of the sun which once grazed upon the plains of Sicily, divided according to color into four herds, one milk-white, one black, one dappled and one yellow. The number of bulls is greater than the number of cows, and the relations between them are as follows:
[c] White bulls = ( 1/2 + 1/3 ) black bulls + yellow bulls,
Black bulls = ( 1/4 + 1/5 ) dappled bulls + yellow bulls,
Dappled bulls = ( 1/6 + 1/7 ) white bulls + yellow bulls,
White cows = ( 1/3 + 1/4 ) black herd,
Black cows = ( 1/4 + 1/5 ) dappled herd,
Dappled cows = ( 1/5 + 1/6 ) yellow herd,
Yellow cows = ( 1/6 + 1/7 ) white herd.[/c]
If thou canst give, O friend, the number of each kind of bulls and cows, thou art no novice in numbers, yet can not be regarded as of high skill. Consider, however, the following additional relations between the bulls of the sun:
[c] White bulls + black bulls = a square number,
Dappled bulls + yellow bulls = a triangular number.
[u]If thou hast computed these also, O friend, and found the total number of cattle,
then exult as a conqueror, for thou hast proved thyself most skilled in numbers.[/u][/c][/color][/i][/b]...................................................................................
The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by Carl Ernst August Amthor (1845–1916), headmaster of the Gymnasium zum Heiligen Kreuz (Gymnasium of the Holy Cross) in Dresden, Germany. Using logarithmic tables, he calculated the first digits of the smallest solution,
[c][size=150][color=#0000FF]showing that it is about 7.76 × 10[sup]206544[/sup] cattle,[/color][/size]
far more than could fit in the observable universe.>>[/c][/quote][quote=https://en.wikipedia.org/wiki/Cattle_of_Helios]
<<In Greek mythology, the Cattle of Helios (Greek: Ἠελίοιο βόες, Hēelíoio bóes), also called the [b][u][color=#0000FF]Oxen of the Sun[/color], are cattle pastured on the island of Thrinacia (sometimes identified with Sicily).[/u][/b]
Helios, also known as the titan of the sun, is said to have had seven herds of oxen and seven flocks of sheep, each numbering fifty head. In the Odyssey, Homer describes these immortal cattle as handsome (ἄριστος), wide-browed, fat (εúρυμέτωπος) and straight-horned (ὀρθόκραιρος). The cattle were guarded by Helios’ daughters, Phaëthusa and Lampetië, and it was known by all that any harm to any single animal was sure to bring down the wrath of the titan.
Tiresias and Circe both warn Odysseus to shun the isle of Helios. When Eurylochus begs to be allowed to land to prepare supper, Odysseus grudgingly agrees on condition that the crew swear that if they come upon a herd of cattle or a great flock of sheep, no one will kill any of them. They are held on the isle for a month by an unfavorable storm sent by Poseidon.
When Odysseus goes up the island to pray to the gods and ask for help, Eurylochus convinces the crew to drive off the best of the cattle of Helios and sacrifice them to the gods: "if he be somewhat wroth for his cattle with straight horns, and is fain to wreck our ship, and the other gods follow his desire, rather with one gulp at the wave would I cast my life away, than be slowly straitened to death in a desert isle." When he returns to the ship, Odysseus rebukes his companions for disobeying his orders. But it is too late, the cattle are dead and gone.
Lampetie tells Helios that Odysseus' men have slain his cattle. In turn, Helios orders the gods to take vengeance on Odysseus' men. He threatens that if they do not pay him full atonement for the cattle, he will take the sun to the Underworld and shine it among the dead. Zeus promises Helios to smite their ship with a lightning bolt and cleave it in pieces in the midst of the ocean.
Soon the gods show signs and wonders to the Odysseus' men. The skins begin creeping and the flesh bellowing upon the spits, both the roast and raw, and there is a sound like the voice of cattle. For six days, Odysseus's company feast on the kine of Helios. On the seventh day, the wind changes. After they set sail, Zeus keeps his word and the ship is destroyed by lightning during a storm and all of his men die. Odysseus escapes by swimming to Calypso's island.>>[/quote]