KANSAS CITY, Mo. — When Caleb Swanigan backs down a defender, the shot clock becomes irrelevant. His options seem unlimited.

In between dribbles, the Purdue star will seamlessly alternate between power and grace, a 6-foot-9, 249-pound forward as comfortable outside as he is around the rim. He makes rebounds appear to be his birthright, makes a behind-the-back pass look like someone six inches smaller threw it.

As a sophomore, Swanigan became the Big Ten Player of the Year, and he is the biggest reason that the fourth-seeded Boilermakers are in the Sweet 16 for the first time in seven years — facing No. 1 Kansas on Thursday night at the Sprint Center.

“You just have to have a resilience about you,” Swanigan said. “That’s the biggest thing, separate off-court stuff from on-court stuff, so it doesn’t translate and make sure you just keep working toward being better every day.”

The youngest of six children, Swanigan was born to a largely absentee father, Carl Sr., who battled a crack-cocaine addiction, and a mother, Tanya, who struggled to maintain employment, subsisting on welfare checks and food stamps.

Routinely shuttled between Utah and Indiana, Swanigan’s address was always temporary, and included stops at various homeless shelters, where he stood on line for food, was surrounded by heavy drug use and relied on donations for much of his wardrobe.

None of his siblings graduated from high school, and three of them — along with his father — spent time in prison. Swanigan never spent more than one year at the same school.

“He appreciates everything,” his adoptive father, Roosevelt Barnes, told The Post. “He knows where he’s been and he doesn’t want to go back there.”

Barnes knew Swanigan back there, a 6-year-old boy in Indiana, who proclaimed he would be “better” than his older brother, Carl Jr. — a one-time Ole Miss recruit who met Barnes through basketball, but lost his right eye in a shooting incident. When Carl Jr. called Barnes in 2011 to see if he would be willing to help and look after his younger brother, Barnes hadn’t seen Swanigan in roughly seven years.

But a few weeks later, Barnes — a former Purdue football/basketball/baseball player who went to the 1980 Final Four and is now a sports agent — brought the 13-year-old to his home in Fort Wayne, Ind., and soon filed adoption papers.

“My kids were grown and gone, I was going through a divorce at the time, and it was a blessing,” Barnes said. “It gave me someone I could love and would be in the house. It gave me someone that I could focus on and someone I knew needed some help.”

Before Swanigan was Indiana’s Mr. Basketball, he was 360 pounds and entering the eighth grade, but even then, Barnes and Swanigan talked about being the “best,” even if it the kid barely could walk a block without needing an oxygen mask.

“Once we made sure he didn’t have any health issues from being obese, and once he decided to be an athlete, what convinced me that he was special was that even though he was extremely heavy, he was really intelligent, and no matter what task I gave him to do, he never quit,” Barnes said. “Most people, when they’re that out of shape, they just give up. It was a tough process. He was willing, but it’s hard to break bad habits. You have to change from the inside out. We didn’t just change his diet, we changed his lifestyle, we changed who he thought he was. He had to believe he could be great. He had to believe he could be the best.”

By the time high school was over, “Biggie” lost more than 100 pounds, and became a McDonald’s High School All-American. Now, at Purdue, he wears the No. 50, in honor of his birth father’s age when he died in 2013, after suffering from diabetes and weighing nearly 500 pounds.

On the day before the biggest game of his career, Swanigan was serious, uninterested in discussing anything but the top-seeded Jayhawks.

“He’s always been goofy, he likes to have a good time, but he won’t let you in unless you really know him,” teammate P.J. Thompson said.

The truth comes out on the court, when Swanigan has the ball and is patiently backing down an opponent. Except now, his options truly are unlimited.

“He’s like a heavyweight boxer that’s going to hit you into your body and break your ribs so that by the end of the game you can’t breathe,” Barnes said. “Most guys don’t have the will to sustain that or to fight that every possession. That’s his approach. It’s not a sexy thing, but that’s how you win championships. That’s how you win in life, really. It’s a day-to-day grind. And he’s winning every day.”