Yamatai was an ancient country somewhere in what is now Japan, described in the ancient Chinese chronicle The History of the Kingdom of Wei, written in A.D. 297. According to the chronicle, Yamatai was a loose confederation of about 30 allied states ruled by a celibate shamaness-queen known as Pimiko. The people of Yamatai are described as exceptionally long-lived, fond of bathing, and enamored of strong liquor - traits remarkably similar to those we find in Japan today.
The exact location of Yamatai has long been disputed by scholars. While the Chinese chronicler kindly left us some very precise directions to get there from China, with a series of instructions on which direction to sail and for how many days, the problem is that if these directions are actually followed exactly, you end up in the middle of the Pacific Ocean.
This has led a few puckish scholars to speculate that Yamatai was actually located in Hawaii! But more serious scholars have speculated that Yamatai must have either been located on the Yamato plain in central Japan, which is where you would come to if you kept the distances the same but not the directions, or else in northern Kyushu, which is were you would get to if you kept the directions the same but assumed that the distances were too large.
The Yamato plain hypothesis is supported by the strong similarity between the the word "Yamato" and the Chinese transliteration "Yamatai," and indeed some scholars have suggested that the three Chinese characters used to write "Yamatai" may well have been pronounced something like "Zhamato" in the 3rd century.
Meanwhile, the northern Kyushu hypothesis is supported by the fact that we have extensive archaelogical evidence of a civilization in northern Kyushu in the 3rd century, whereas the Yamato plain seems to have been less developed at this point. There is also evidence that this northern Kyushu civilization may have moved to what is now called the Yamato plain a few centuries later, and conceivably could have carried the name "Yamato" with them.