One of the most common ways for chess historians to trace when the board game chess entered a country is to look at the literature of that country. Although due to the names associated with chess sometimes being used for more then one game (for instance Xiang-qi in China and Tables in England), the only certain reference to chess is often several hundred years later than uncertain earlier references.
The earliest dates for strong references include,
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a. 923 AD - at-Tabari's Kitab akhbar ar-rusul wal-muluk (note the work is an arabic work, no early greek works are known)
c. 900 AD - Huan Kwai Lu ('Book of Marvels')
c. 1180 AD - Alexander Neckam's De Natura Rerum (note that it is thought that Neckam may have learnt of chess in Italy, not in England)
a. 1127 AD - A song of Guilhem IX Count of Poitiers and Duke of Aquitaine.
c. 1030 AD - Ruodlieb
1148 AD - Kalhana's Rajatarangini (translated by MA Stein, 1900)
The King, though he had taken two kings (Lothana and Vigraharaja) was helpless and perplexed about the attack on the remaining one, just as a player of chess (who has taken two Kings and is perplexed about taking a third).
(note this refers to the old four-handed chess sometime known as chaturagi).
c. 1062 AD - Letter from Petrus Damiani (Cardinal Bishop of Ostia) to the Pope-elect Alexander II and the Archdeacon Hildebrand.
c. 600 AD - Karnamak-i-Artakhshatr-i-Papakan
Artakhshir did this, and by God's help he became doughtier and more skilled than them all in ball-play, in horsemanship, in chess, in hunting and in all other accomplishments.
(It is fairly certain chess is meant due to the word chatrang being used).
c. 1009 AD - castrensian will of Ermengaud I (Count of Urgel)
I order you, my executors, to give . . . these my chessmen to the convent of St. Giles, for the work of the church.
c. 1620 AD - Sejarah Malayu
Now this Tan Bahra was a very skillful chessplayer, and one that was unequalled at the game in that age, and he played at chess with the men of Malacca.
c. 1000 AD - Manuscript 319 at Stiftsbibliothek Einsiedeln.
HJR Murray, A History of Chess, (Oxford University Press)
Helena M. Gamer, The Earliest Evidence of Chess in Western Literature: The Einsiedeln Verses, Speculum, Vol. 29, No. 4. (October 1954), pp. 734-750.