N.H. Rev. Stat. Ann. § 519:19

Each party in a civil cause is entitled to three peremptory challenges of jurors drawn for the trial. The peremptory challenges shall be exercised alternately, beginning with the plaintiff. When the plaintiff and defendant have exhausted their challenges the court shall take unusual care that the jurors who take the places of the ones last challenged shall be wholly indifferent and impartial and reasonably satisfactory to both sides.