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With that in mind, here's a look at the top player drafted by the Twins in each of the top 15 rounds of the Draft.
Round 1: Joe Mauer, 2001
The Twins went with a hometown selection for the first overall pick of the 2001 Draft, taking St. Paul's Mauer over Mark Prior. The decision more than worked out for the Twins, as Mauer has been a six-time All-Star and won the American League MVP Award in 2009. The Twins have had plenty of success with first-rounders, including Torii Hunter, Chuck Knoblauch, Denard Span, Michael Cuddyer, Matt Garza and Glen Perkins. It's worth noting that Hall of Famer Kirby Puckett was a first-rounder, but in the former January Draft that was eliminated in 1986.
Round 2: Frank Viola, 1981
Viola pitched eight seasons in Minnesota, posting a 3.86 ERA and winning 112 games. He also won the AL Cy Young Award in '88, a year after helping the Twins to their first-ever World Series title in '87. Viola was named World Series MVP in '87.
Round 3: Bert Blyleven, 1969
Blyleven was one of the best pitchers of his generation, and he was inducted into the National Baseball Hall of Fame in 2011. The right-hander pitched with the Twins from 1970-76, and again from 1985-88 as part of a career that spanned 22 seasons. He also helped the Twins to the World Series in '87. Other notable third-rounders include Justin Morneau and A.J. Pierzynski.
Round 4: Graig Nettles, 1965
Nettles only played three seasons with the Twins before being traded to the Indians before the 1970 season. He went on to a solid career, as he was considered one of the best defensive third basemen of his time and was a six-time All-Star. He also won World Series titles with the Yankees in '77 and '78.
Round 5: Dave Goltz, 1967
The fifth round hasn't been kind to the Twins, as Goltz and Doug Mientkiewicz are the only two big league regulars the organization has ever selected in that round. Goltz pitched with the Twins from 1972-79, posting a 3.48 ERA. He was a 20-game winner in '77 and finished sixth in the balloting for the Cy Young Award that year.
Round 6: Pat Neshek, 2002
Neshek is the best of a mostly uninspiring group of players taken by the Twins in the sixth round. Neshek, a fan-favorite during his time in Minnesota, developed into a solid reliever with the Twins. He had two strong seasons in '06 and '07 before he was placed on waivers before the '11 season.
Round 7: LaTroy Hawkins, 1991
Hawkins was drafted 23 years ago but is still in the Majors, serving as closer for the Rockies. He played with the Twins from 1995-2003, and has played for nine teams since. He started his career as a starter but became a reliever in 2000.
Round 8: Brad Radke, 1991
Radke is one of the more celebrated pitchers in Twins history, as he played his entire career in Minnesota from 1995-2006. He finished third in the balloting for AL Cy Young Award in '97 and was an All-Star in '98. He posted a 4.22 ERA with 148 wins in his career.
Round 9: Steve Lombardozzi, 1981
The Twins haven't had much success this round, but Lombardozzi was a solid defender at second base and his defense helped Minnesota to the title in '87. But he hit just .233 in his four years with the Twins, as his offense never could match his defense.
Round 10: Steve Braun, 1966
Braun missed the first two years of his career as he was on baseball's military list, but he made his Twins debut in '71. He played six years with the Twins, hitting .284 with 35 homers and 32 stolen bases in 751 games. He played for four more teams in a career that lasted from 1971-85.
Round 11: None
The Twins have had little luck in the 11th round of the Draft, as the only big league regular drafted and signed by Minnesota that round is reliever Evan Meek, who was an All-Star with the Pirates in 2010. But the Twins released Meek before he ever got to the big leagues.
Round 12: Jason Kubel, 2000
Kubel was one of baseball's top prospects but suffered a gruesome knee injury while playing in the Arizona Fall League in 2004 that forced him to miss all of the '05 season. But he bounced back to have a nice career as a corner outfielder/designated hitter. He played with the Twins from 2004-11 before rejoining Minnesota this season.
Round 13: Matt Lawton, 1991
Lawton was a solid outfielder for the Twins from 1995-2001, hitting .277/.379/.428 in seven seasons in Minnesota. He was named an All-Star with the Twins in '00. He played 12 years in the Majors with seven different teams.
Round 14: Mike Trombley, 1989
Trombley pitched with the Twins from 1992-99, making 365 career appearances, including 36 starts. He had a 4.53 ERA in his nine years in Minnesota.
Round 15: Rick Dempsey, 1967
Dempsey went on to be one of the best defensive catchers of his era but never really got a shot with the Twins from 1969-72, as he played in just 41 games in the Majors over those four years. Dempsey went on to play 24 years in the Majors, winning the World Series in '83 with the Orioles and '88 with the Dodgers.