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L _ G H T _ R Clue
A MINIMUM OF 3 WEIGHINGS IS REQUIRED. DIVIDE THE 27 BALLS INTO 3 GROUPS OF 9 BALLS EACH. USE THE SCALE TO WEIGH 2 GROUPS. THIS WILL TELL US WHICH GROUP HAS THE HEAVIER BALL. NOW, DIVIDE THE 9 BALLS INTO 3 GROUPS OF 3 BALLS EACH AND WEIGH 2 GROUPS, AND IDENTIFY THE GROUP WHICH HAS THE HEAVIER BALL. FROM THE HEAVIER GROUP, WEIGH 2 BALLS TO DETERMINE THE HEAVIER BALL STEP 1: DIVIDE IT INTO SETS OF THREE. IF YOU PUT ONE SET ON EITHER SIDE, IT'LL EITHER EQUAL THE SAME, OR YOU'LL SEE WHICH SET IS LIGHTER. EITHER WAY, YOU NARROW IT DOWN TO THREE BALLS. STEP 2: PUT TWO OF THE BALLS ON THE SCALE. IF THEY'RE EQUAL, THE LAST BALL IS DEFECTIVE, BUT IF NOT, THE SCALE WILL TELL YOU WHICH ONE IS LIGHTER WHEN THE FIRST SERVANT COMES IN, THE KING SHOULD WRITE DOWN HIS NUMBER. FOR EACH OTHER SERVANT THAT REPORTS IN, THE KING SHOULD ADD THAT SERVANT'S NUMBER TO THE CURRENT NUMBER WRITTEN ON THE PAPER, AND THEN WRITE THIS NEW NUMBER ON THE PAPER. LET X BE THE NUMBER OF THE MISSING SERVANT AND Y BE THE NUMBER THAT THE KING HAS WRITTEN. ONCE THE FINAL SERVANT HAS REPORTED IN, THE NUMBER ON THE PAPER SHOULD EQUAL: Y = (1 + 2 + 3 + ... + 99 + 100) - X (1 + 2 + 3 + ... + 99 + 100) = 5050, SO WE CAN REPHRASE THIS TO SAY THAT THE NUMBER ON THE PAPER SHOULD EQUAL: Y = 5050 - X SO TO FIGURE OUT THE MISSING SERVANT'S NUMBER, THE KING SIMPLY NEEDS TO SUBTRACT THE NUMBER WRITTEN ON HIS PAPER FROM 5050: 5050 - Y = X TAKE 6 BALLS AND SPLIT THEM INTO 2 GROUPS OF 3 BALLS EACH. USE THE SCALE TO WEIGH THE 2 GROUPS. IF ONE OF THE GROUPS IS HEAVIER THAN THE OTHER GROUP, PICK 2 BALLS FROM THE HEAVIER GROUP AND USE THE SCALE TO WEIGH THEM. IF THEY ARE EQUAL IN WEIGHT, THE OTHER BALL IN THAT GROUP IS THE HEAVY ONE. IF BOTH THE GROUPS ARE EQUAL IN WEIGHT, THEN PICK THE REMAINING 2 BALLS AND USE THE SCALE TO DETERMINE THE HEAVIER BALL