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22 TIMES, NOT 24 TIMES. IN T HOURS, THE MINUTE HAND COMPLETES T LAPS. IN THE SAME AMOUNT OF TIME, THE HOUR HAND COMPLETES T/12 LAPS. THE FIRST TIME THE MINUTE AND HOUR HANDS OVERLAP, THE MINUTE HAND WOULD HAVE COMPLETED 1 LAP MORE THAN THE HOUR HAND. SO WE HAVE T = T/12 + 1. THIS IMPLIES THAT THE FIRST OVERLAP HAPPENS AFTER T = 12/11 HOURS (~1:05 AM). SIMILARLY, THE SECOND TIME THEY OVERLAP, THE MINUTE HAND WOULD HAVE COMPLETED TWO MORE LAPS THAN THE HOUR HAND. SO FOR N OVERLAPS, WE HAVE T = T/12 + N. SINCE WE HAVE 24 HOURS IN A DAY, WE CAN SOLVE THE ABOVE EQUATION FOR N: 24 = 24/12 + N 24 = 2 + N N = 22 THE HANDS WILL OVERLAP AT 12:00, 1:05, 2:10, 3:15, 4:20, 5:25, 6:30, 7:35, 8:40, 9:45, AND 10:50. DO CONSIDER THE FACT THAT THERE WILL BE NO 11:55. IT BECOMES 12:00 THE MINIMUM NUMBER OF WEIGHTS REQUIRED IS FIVE AND THESE SHOULD WEIGHT 1, 3, 9, 27 AND 81 POUNDS. THE MERCHANT HAS TO USE A BALANCE WEIGHING SCALE TO DO THE JOB. TO WEIGH 2 POUNDS, HE'LL HAVE TO PUT THE 3 POUND WEIGHT ON ONE PAN AND 1 POUND WEIGHT ON THE OTHER PAN. TO WEIGH 5 POUNDS, HE'LL HAVE TO PUT THE 9 POUND WEIGHT ON ONE PAN AND 1 AND 3 POUND WEIGHTS ON THE OTHER PAN THE MERCHANT HAS 32 GOLD COINS. TO VERIFY THIS, DIVIDE THE 32 COINS INTO TWO UNEQUAL NUMBERS, SAY, 27 AND 5. THEN: 32 (27 - 5) = (272) - (52). 704 = 729 - 25 IF THE TWO UNEQUAL NUMBERS ARE 22 AND 10, THEN: 32 (22 - 10) = (222) - (102). 384 = 484 - 100 ALTERNATIVELY, LET'S SAY THAT THE 2 NUMBERS ARE X AND Y. WE CAN THEN COME UP WITH THE FOLLOWING EQUATION: 32(X - Y) = X2 - Y2 X2 - Y2 CAN BE EXPANDED TO: (X - Y)(X + Y) SO, NOW THE EQUATION WILL BECOME: 32 = X + Y (X + Y) WILL GIVE THE TOTAL NUMBER OF GOLD COINS IT WILL TAKE 198 DAYS. LET'S SAY THE TREE WAS ORIGINALLY ONE FOOT TALL. ON THE FIRST DAY, IT GREW HALF A FOOT; ON THE NEXT DAY 1⁄3 OF 11⁄2 FEET, WHICH IS HALF A FOOT; AND SO ON - IT GREW HALF A FOOT EVERY DAY. SO IN 198 DAYS, IT GREW 99 FEET, AND THEN WAS 100 FEET TALL