In all cases, when subsequent to the time provided by law for persons to claim lots on such townsites, and prior to the taking effect of this law, any person may have entered thereon and improved any lots belonging to such town, such person, after the report of such board of appraisers, and prior to the public sale, may purchase any such lots from the said trustees or commissioners at private sale, for cash, at the appraised value of such lots, exclusive of improvements, unless there shall be adverse claimants to any such lots, in which case the respective rights of such claimants shall be determined, as hereinafter provided in this chapter.