The U.S. invented basketball but does not dominate one form, three-on-three, when it comes to Olympics qualifying. So, go … Latvia?When three-on-three basketball has its Olympics debut in Tokyo this summer, Latvia will be there. So will Poland. But a men’s team from the United States, birthplace of basketball and winner of 15 of 19 men’s gold medals in the traditional five-on-five game, will not.The Americans were eliminated from a qualifying tournament in Austria on Sunday, 21-16, by the Netherlands.How did this happen?One clue is the American men’s roster. It consisted of Dominique Jones, Robbie Hummel, Kareem Maddox and Joey King. All were capable college players. All have honed their games in three-on-three play for several years in preparation for the Games. But they hardly represent the best players the U.S. could offer.Several factors prevented the U.S. from rolling out a threesome like LeBron James, Stephen Curry and Kevin Durant. For one, the best N.B.A. players are currently in the middle of the playoffs.Still, even hustling together a group of stars from teams not in the playoffs would not have been possible. Hoping to promote its three-on-three tournaments around the world, the international federation, FIBA, mandated that Olympic hopefuls earn points in those events. Since they took place in the winter, all N.B.A. players and even most Americans playing for European teams were out.FIBA is aware of what it is doing. It hopes to build the game beyond merely the Olympics, so it demands participation in its other, lower-profile events. It also wants to have as many countries involved as possible. The U.S. men did not qualify for the 2018 World Cup either, though Andorra, population 77,000, did, in part merely for showing up.“Andorra has heavy participation every weekend,” an official explained at the time.The unforgiving nature of the qualifying tournament over the weekend didn’t help. The U.S. won its pool with a 3-1 record. But that only got them into the quarterfinals. The loss to the Netherlands eliminated them, since only the top three teams advanced to the eight-team Olympic tournament.Dominique Jones was on the U.S. three-on-three roster but it lacked the biggest names. Christian Ugarte/EPA, via ShutterstockIn the three-on-three game — or 3×3 as FIBA insists on calling it — baskets are worth 1 point. What would be a 3-point shot in the regular game is worth 2 points. The Dutch made four 2-pointers down the stretch to put the Americans away.“Obviously, it’s not the ending we wanted,” said Joe Lewandowski, who has the title of coach/adviser (coaching is not allowed during games). He is the head coach at Point Park University, an N.A.I.A. school in Pittsburgh. The traditional men’s five-on-five team has qualified and will be coached by Gregg Popovich of the San Antonio Spurs.Another American team in a very American sport will face its own challenge this week when men’s baseball sends minor leaguers to try to qualify for the Games.Kara Lawson of Duke coached the American women’s three-on-three team, which qualified for the Games with ease, finishing 4-0 in its pool, then beating Belarus, 21-10, and Spain 21-13. The winner is the first team to 21 points or the leader after 10 minutes. (Dawn Staley of South Carolina will coach the women’s five-on-five at the Games.)The women’s three-on-three roster has much more familiar names, all with W.N.B.A. experience: Allisha Gray of the Dallas Wings, Stefanie Dolson of the Chicago Sky, Katie Lou Samuelson of the Seattle Storm and Kelsey Plum of the Las Vegas Aces, a former No. 1 overall draft pick. Because the W.N.B.A. is a summer league, the opportunity was there to recruit some top players who were available to earn points at FIBA events in the winter.Those stars will now return to their W.N.B.A. teams. In contrast, the players on the men’s teams include a news producer and a development specialist.Though one spot is left in the men’s field at the Olympics, the U.S. is not eligible for it. In another quirky FIBA decision, that place is reserved only for nations that did not have a team in the 2012 or 2016 Olympic Games. So men from Mongolia, Hungary or Romania could be at the Games, while the nation of Wilt Chamberlain and Julius Erving watches from home. More