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Book of Mormon/Translation/Method/Did Joseph place the Urim and Thummim in his hat
Joseph Smith placed either the interpreters or the seer stone in a hat
Gospel Topics: "Joseph placed either the interpreters or the seer stone in a hat"
"Book of Mormon Translation," Gospel Topics on LDS.org:
According to these accounts, Joseph placed either the interpreters or the seer stone in a hat, pressed his face into the hat to block out extraneous light, and read aloud the English words that appeared on the instrument. 
Question: What are the Nephite interpreters?
The Nephite interpreters are two seer stones set in a framework resembling a set of "spectacles"
The Lord provided a set of seer stones (which were formerly used by Nephite prophets) along with the plates. The term Nephite interpreters can alternatively refer to the stones themselves or the stones in conjunction with their associated paraphernalia (holding rim and breastplate). Some time after the translation, early saints noticed similarities with the seer stones and related paraphernalia used by High Priests in the Old Testament and began to use the term Urim and Thummim interchangeably with the Nephite interpreters and Joseph's other seer stones as well. The now popular use of the term Urim and Thummim has unfortunately obscured the fact that all such devices belong in the same class of consecrated revelatory aids and that more than one were used in the translation.
The manner in which the interpreters were used was never explained in detail
The Nephite interpreters were intended to assist Joseph in the initial translation process, yet the manner in which they were employed was never explained in detail. The fact that the Nephite interpreters were set in rims resembling a pair of spectacles has led some to believe that they may have been worn like a pair of glasses, with Joseph viewing the characters on the plates through them. This, however, is merely speculation that doesn't take into account that Joseph soon disassembled the fixture, the spacing between seer stones being too wide for his eyes. The accompanying breastplate also appeared to have been used by a larger man. Like its biblical counterpart (the High Priest's breastplate contained 12 gems that symbolized him acting as a mediator between God and Israel), the Nephite breastplate was apparently non-essential to the revelatory process.
Question: Did Joseph ever place the Nephite interpreters ("Urim and Thummim") into his hat?
There is evidence that indicates that Joseph did place the "Urim and Thummim" into his hat
The Church states that, "These two instruments—the interpreters and the seer stone—were apparently interchangeable and worked in much the same way such that, in the course of time, Joseph Smith and his associates often used the term “Urim and Thummim” to refer to the single stone as well as the interpreters." and "According to these accounts, Joseph placed either the interpreters or the seer stone in a hat, pressed his face into the hat to block out extraneous light, and read aloud the English words that appeared on the instrument." 
Contemporary accounts indicate that Joseph began the translation using the Nephite interpreters, and finished it using his own seer stone after the loss of the 116 pages of manuscript. Note this account from Martin Harris:
The two stones set in a bow of silver were about two inches in diameter, perfectly round, and about five-eighths of an inch thick at the centre; but not so thick at the edges where they came into the bow. They were joined by a round bar of silver, about three-eighths of an inch in diameter, and about four inches long, which, with the two stones, would make eight inches. The stones were white, like polished marble, with a few gray streaks. I never dared to look into them by placing them in the hat, because Moses said that “no man could see God and live,” and we could see anything we wished by looking into them; and I could not keep the desire to see God out of my mind. And beside, we had a command to let no man look into them, except by the command of God, lest he should “look aught and perish.” 
Martin Harris said that Joseph placed the interpreters in a hat
Harris states that Joseph used the "two stones set in a bow of silver" by "placing them in the hat." He is referring to the Nephite interpreters, what we today refer to as the "Urim and Thummim". Joseph may have therefore placed the Nephite interpreters into his hat - a method of receiving revelation that he was already quite familiar with.
Based upon these accounts, it appears that Joseph began the translation process using the Nephite interpreters, and that at some point he may have used them with a hat. After the loss of the 116 pages, he may have either switched to his own seer stone or continued to use the Nephite “spectacles,” again with the hat. In fact, given the consistent reports of the use of the hat during translation, it is not possible to know with certainty whether Joseph was using the Nephite interpreters or the seer stone in the hat during this period of time. One thing seems certain based upon witness accounts—during the period of the translation process after the loss of the 116 pages, Joseph sat in the open, without a curtain, dictating to his scribe while looking into his hat.
- "Book of Mormon Translation," Gospel Topics on LDS.org (2013)
- "Book of Mormon Translation," Gospel Topics on LDS.org
- “Martin Harris Interview with Joel Tiffany, 1859,” in Early Mormon Documents, 2:305.
- Roger Nicholson, "The Spectacles, the Stone, the Hat, and the Book: A Twenty-first Century Believer’s View of the Book of Mormon Translation," Interpreter: A Journal of Mormon Scripture 5 (2013): 121-190