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THE MAN IS A DWARF. HE CAN'T REACH THE UPPER ELEVATOR BUTTONS, BUT HE CAN ASK PEOPLE TO PUSH THEM FOR HIM. IF IT IS RAINING, HE PUSHES THE BUTTONS WITH HIS UMBRELLA 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 WHEN THE PROFESSOR CONFIRMED JOHNNY'S ANONYMITY (THE PROFESSOR DIDN'T KNOW WHO HE WAS), JOHNNY QUICKLY SLIPPED HIS EXAM PAPER INTO THE PILE OF EXAMS AND RAN OFF, SO THE PROFESSOR WOULDN'T KNOW WHICH EXAM PAPER DESERVED AN F. CONGRATULATIONS ON STICKING THROUGH THE WHOLE RIDDLE HE WORKS IN THE ENGINE ROOM OF A LINER! TO GET TO WORK, HE WALKS ALONG THE DECKS FROM HIS CABIN FACING FORWARDS, AND DOWN THE LADDERS BETWEEN DECKS FACING BACKWARDS. HOWEVER, WHEN HE FINISHES, HE ONLY NEEDS TO FACE FORWARDS TO CLIMB THE LADDERS AGAIN AND WALK ALONG THE DECK BACK TO HIS CABIN