Before departing for the Algarve Cup, United States coach Greg Ryan offered a best-case scenario for his team's stay in Portugal.
"We're always hoping we can get a matchup with Germany over there," Ryan said. "We've been fortunate the last two years to be able to play them in the final, and we're hoping we can get there again."
But as anyone who survived rather than thrived in high school algebra can attest, equations with multiple variables can be tricky.
The United States will indeed play for the championship on Wednesday, but Germany is nowhere to be seen after a surprisingly poor performance from the reigning World Cup champions just six months before they will try to defend that title in China. As a result, Ryan won't have a chance to seek revenge for last year's penalty-kick shootout against the Germans, the only game in which the United States has walked off the field a loser (although it went as a draw in the official records) since he took over prior to the 2005 Algarve Cup.
Ryan's team earned a spot opposite Denmark in the final on the strength of a 3-2 win against Sweden in the final game of group play. Despite winning both of its first two matches, 2-1 against China and 1-0 against Finland, the Americans needed an outright win against the Swedes to avoid losing out on goal differential.
That they will face a third Scandinavian side, especially one other than Norway, is something of a shock.
Just one year removed from losing handily against Germany and China in group play at this event, Denmark reached the final on the strength of four goals in an opening win against France, a subsequent 1-0 upset of Norway and the war of attrition that took place in a group that also included Germany.
By the time the Danes squared off against the Germans on Monday, their place in the championship game was all but assured. Even the resulting 3-0 loss left them well clear of runner-up France in the final standings for Group B.
Although Denmark has played previously for the title in the Algarve, losing to Sweden in 2001, this year's run is a dramatic reversal of fortune for a team that finished ninth last year and hadn't fared better than sixth since 2002. It's also something of a confirmation of the team's solid run in World Cup qualifying, which saw the Danes score 22 goals and concede just six on their way to 19 points and a Group 3 win over main rivals Finland and Spain.
Their recent history against the United States is a similarly mixed bag. Denmark tied the United States 1-1 in the Peace Queen Cup last November in South Korea, earning the point despite being outshot 16-5 and conceding 11 corner kicks against just one earned. That result stood in stark contrast to last year's Algarve, when the United States scored three goals in the opening 31 minutes and cruised to a 5-0 victory in group play.
Jenni Branam started in goal for Ryan in that game, marking the last time a player other than Hope Solo or Brianna Scurry appeared in net. Presumably, Solo will start her third game of the tournament against Denmark, although Scurry has the team's only clean sheet after making two saves in the win over Finland.
Without a chance to measure his team against Germany, Ryan may opt to continue mixing and matching his lineup as he aims to measure his team's development as much as he strategizes for the opponent. Only defenders Cat Whitehill and Christie Rampone started all three games in group play, despite an extra day of rest between the second and third games.
And while finding offense has been a theme for Ryan almost since the day he took over, it may be the defense that's in the spotlight against Denmark. The three goals scored in the tournament against the United States are the most in a three-game span since late spring of last year, when the United States sandwiched a 3-1 win against Japan and a 3-2 win against Sweden around a 1-0 win against Japan. Even though one of Sweden's goals came on a late penalty kick, that's not a trend the back line will want to bring back across the Atlantic.
Making marking matters more challenging, each of Denmark's five goals in group play was scored by a different player.
"They are a very attack-oriented team," Ryan told U.S. Soccer's Web site. "They send five players on your restraining line and so it's just going to be a game of attacking. We're not going sit back and defend them all day. We're going to get numbers forward and go after them. It could be a very wide-open game."
Defender India Trotter is one of just two players, along with midfielder Marci Jobson (formerly Marci Miller), not to play in any of the first three games. Although giving Trotter a prominent role in the title game might be an unorthodox way to break her in, it's difficult to imagine Ryan bringing her to Portugal without getting her at least a half of game competition, especially given her offensive background as he seeks to push the attack.
The win against Sweden also marked the first time in the tournament that Ryan paired Shannon Boxx and Leslie Osborne together in midfield, with Boxx coming on in place of Lori Chalupny in the 63rd minute. The results looked nice on the score sheet, with Boxx setting up what proved to be the game-winning goal on a header by Abby Wambach.
Osborne, of course, rose to prominence filling in for Boxx at holding midfielder in the second half of last year, but with Boxx regaining her form, it will be incumbent on Ryan to find ways to get his potentially two most consistent midfield players on the field as much as possible.
Ultimately, this may not be the test Ryan had hoped for as a reward for the United States taking care of business in group play, even if Denmark looks increasingly like a worthy sleeper. But Germany will have to wait.
Graham Hays is a regular contributor to ESPN.com's soccer coverage. E-mail him at Graham.Hays@espn3.com.