On April 1 in each year, the Treasurer shall make up a list of all delinquent taxpayers and the amounts for which they are respectively in arrears and publish the same during each of two (2) successive weeks before the first Monday in May following. If such taxes and costs are unpaid on that day, he shall immediately thereafter make up a list of all delinquents assessed with real estate, with a notice appended that if such taxes, interest, costs, expenses and fees are not paid on or before the third Monday in June next ensuing, the Treasurer will proceed at 10:00 a.m. on that day, at a place designated by him, to offer such property for sale to the highest bidder for cash. Such list and notice shall be published at least four (4) weeks prior to the third Monday in June. Upon the third Monday in June in each year at 10:00 a.m., the Treasurer shall proceed to sell, under the terms of such notice, all property upon which taxes, interest, costs, expenses and fees are in arrears and shall continue such sale from day to day on each secular day, legal holidays excepted, from 10:00 a.m. to 5:00 p.m. until he shall have offered and disposed of all of such property.