Chris Chambliss, center, and Roy White, right, greet Bucky Dent after his three-run homer gave the Yankees a 3-2 lead during a one-game playoff on Oct. 2, 1978, at Fenway Park. New York went on to beat the Red Sox, 5-4, to capture the American League East title. Michael Maher photo

The Boston Red Sox and New York Yankees were born to be enemies.

When American League founder Ban Johnson decided he wanted a team in New York, he funneled several top players, some from Boston, to the new franchise in 1903, fostering resentment.

Still the Red Sox had the upper hand for 20 years before selling Babe Ruth to the Yankees in 1920. And whether or not one believes in “curses,” the 20th century belonged to New York, with 26 championships to Boston’s five.

The 21st century has been a different story, with Boston leading in championships 4-1.

But as the franchises’ fortunes see-sawed, the intense rivalry between the teams has remained a constant. Rarely has one team reached the championship without going through the other.

On Tuesday night, the Yankees and Red Sox, tied at 92-70, will meet at Fenway Park in the American League wild-card game. The winner advances to the AL Division Series. It will be the sixth time in their histories that the season will hinge on a single day.

Here is how the first five winner-take-all matchups turned out:

Oct. 10, 1904: One that slipped away

The Boston Americans (now Red Sox) won the first World Series in 1903, then the New York Highlanders (now Yankees) challenged in 1904. Midseason, New York ace Jack Chesbro unveiled a spitball and became unhittable, winning 41 games in a record that still stands. But then “Happy Jack,” at his own insistence, pitched too much down the stretch.

On Oct. 10, the teams met in a doubleheader to decide the pennant with New York needing to win both. Boston clinched with a 3-2 win in the first game, after Lou Criger scored the go-ahead run on a wild pitch. One of Chesbro’s slippery offerings had gotten away, and so had the pennant. There was no World Series in 1904 when the National League champion New York Giants refused to play against Boston.

Chesbro was never the same pitcher and was haunted by the wild pitch the rest of his life, maintaining it should have been caught by his catcher.

“I would have given my entire year’s salary back could I have had that ball back,” Chesbro later said.

Oct. 2, 1949: Down to the wire

Dormant for decades, the Red Sox challenged Yankees supremacy in the years following World War II. The teams engaged in a tense race throughout the 1949 season, with the Red Sox leading by one game with two to play, and those two were in Yankee Stadium. There was no divisional play then, the winner went right to the World Series.

The Yankees stayed alive Oct. 1 when Johnny Lindell, a little-known pitcher-turned-outfielder, homered in the ninth inning. On Oct. 2, with 68,000 jammed into the Bronx ballpark, the Yankees broke open a 1-0 game with four runs in the eighth inning, then held on to win, 5-3, behind Vic Raschi, a righthander from Western Massachusetts known as “The Springfield Rifle.”

The Yankees went on to win the World Series, their first of five in a row.

Oct. 2, 1978: Bucky Dent’s Monster HR

Russell Earl Dent, the light-hitting shortstop, is better known as “Bucky,” and Earl was his middle name, not the one Red Sox fans gave him after his three-run homer turned around one of the most memorable games in baseball history. The Yankees, who had trailed by as much as 14 games in mid-July, surged and caught the Red Sox with a stunning four-game sweep at Fenway in early September. Then the Red Sox won their final eight games to catch the Yankees from behind.

They tied for the AL East title, and a 163rd game was needed. The Red Sox led 2-0 until Dent, breaking his bat in the seventh inning, borrowed one from Mickey Rivers and hit a three-run homer that just made it over The Green Monster in the seventh, only his fifth homer of the year. The Yankees clung to their lead, winning 5-4 when Carl Yastrzemski popped up to end the game with the tying and winning runs on base.

The Yankees went on to win the 1978 World Series, repeating as champs. Dent was the World Series’ MVP.

Oct. 16, 2003: Words of wisdom

The Babe Ruth curse was still being talked about as the Red Sox were trying to end their 85-year drought without a championship. Now, divisional play allowed both the Red Sox and Yankees to make the playoffs, and they met for the second time in the American League Championship Series. The Yankees won the 1999 ALCS in five games. This time, the Red Sox forced a seventh game in a series marked by brawls. Red Sox Manager Grady Little suggested it had been elevated to “a war.”

The Red Sox led 4-0 in Game 7 in New York, and Little tried to get one inning too many from ace Pedro Martinez, a move that cost him his job. The Yankees rallied to tie the game, and on it went into extra innings.

By the 11th, Aaron Boone, benched because he had been in a terrible slump, was in the game and about to face knuckleballer Tim Wakefield. As he left the dugout, Yankees Manager Joe Torre gave Boone a piece of hitting advice Red Sox legend Ted Williams had once given him. Torre told Boone to try to hit a line drive to the second baseman, and this would help him keep his weight back and prevent him swinging too early. Boone, now the Yankees’ manager, waited on a floating knuckler and hit it into the left-field seats to send the Yankees to the World Series, which they lost to the Florida Marlins.

After the game, Yankees pitchers Roger Clemens and David Wells, and pitching coach Mel Stottlemyre went to Monument Park for a champagne toast with Babe Ruth’s monument.

Oct 20, 2004: Always a first time

After past postseason failures, the Red Sox were known to disappear for a few years. This time, they shrugged off Boone’s homer and returned to the ALCS to face the Yankees in 2004. Boston was the hotter team and favored to win, but the Yankees won the first three games, including a Game 3 19-8 drubbing.

The Red Sox scrapped to stay alive and won the next three tight games – two on clutch David Ortiz hits – to tie the series. Bidding become the first team ever to overcome a 3-0 deficit in a seven-game series, the Red Sox won in anticlimactic fashion, a 10-3 blowout in which Ortiz hit a two-run homer and Johnny Damon hit two, including a grand slam.

As the Red Sox partied on the field at Yankee Stadium, Yankees owner George Steinbrenner reportedly said, “let them celebrate, they’ve earned it.”

The New York Daily News featured Martinez on its back page with the headline “Hell Freezes Over.” The front page headline read, “The Choke’s On Us.” The Red Sox went on to win the World Series for the first time since 1918.