For N.F.L. Perfection, a Steep Price

Allie and Austin Kiick holding an urn with ashes of their father, Jim, a member of the undefeated 1972 Miami Dolphins team who died in 2020.Credit...Maggie Steber for The New York TimesFORT LAUDERDALE, Fla. — On a sunny January day nearly 50 years ago, Nick Buoniconti, Jim Kiick and Jake Scott ran off the field at the Los Angeles Memorial Coliseum not just as Super Bowl champions, but as part of the only team in N.F.L. history to complete a perfect season.

They all played pivotal roles in the 1972 Miami Dolphins’ final victory. Scott, the gifted free safety, was named most valuable player of that game. Buoniconti, the pugnacious leader of the team’s No-Name Defense, made a key interception.…