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9 Clue
THE PROFESSOR HAS TO ADD THE REST OF THE DIGITS, FIND THE NEAREST NUMBER TO THE SUM THAT IS DIVISIBLE BY 9 AND GET THE DIFFERENCE. SO, JOHN GAVE THE NUMBER 9646 TO THE PROFESSOR. THE PROFESSOR WILL ADD THE NUMBERS (9 + 6 + 4 + 6) TO GET 25. THE NEAREST NUMBER TO 25 THAT IS DIVISIBLE BY 9 IS 27. AND THE CROSSED OUT NUMBER IS 27 - 25. THIS IS A MATHS TRICK THAT RELIES ON THE POWER OF 9 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 THE PROBLEM IS THAT THE QUESTION IS CLEVERLY PHRASED TO CONCEAL WHAT IS REALLY GOING ON. LET'S LOCATE ALL THAT MONEY. THERE ARE TWO WAYS TO THINK ABOUT HOW MUCH MONEY IS OUT THERE TO BE FOUND.THE WAY THAT THIS QUESTION IS TRICKY IS THAT IT COMBINES THAT TWO WAYS. THE FIRST WAY IS THIS. HOW MUCH MONEY DID THE THREE MEN PAY ORIGINALLY? THE SECOND WAY IS THIS. HOW MUCH MONEY DID THEY END UP PAYING? SO IF IT IS THE FIRST WAY, THEN CLEARLY THE TOTAL WE NEED TO ACCOUNT FOR IS THE $15. SO LET'S SEE WHAT HAPPENS TO THAT $15. THE CHEF GETS $10, THE WAITER GETS $2 AND THE GUYS GET $3 BACK. THAT ADDS UP FINE. NOW LET'S LOOK AT THE SECOND WAY. HOW MUCH MONEY DID THEY END UP PAYING? WELL $12, AND $10 OF IT WENT TO THE CHEF AND $2 TO THE WAITER. NOW, THAT ADDS UP TOO. THE PROBLEM WITH THE QUESTION IS THAT THE THE $2 THAT THE WAITER TOOK IS CONTAINED IN THE $12 THAT THEY END UP PAYING SO WE SHOULDN'T EXPECT THEM TO ADD TO ANYTHING MEANINGFUL IF YOU FOLLOWED ALL THE STEPS APART FROM THE LAST ONE THERE WILL BE THREE OPTIONS REMAINING: 64, 84, AND 86. YOU THEN HAD TO ADD UP THE DIGITS, 64=6+4=10, 84=8+4=12, AND 86=8+6=14. FINALLY, YOU THEN HAD TO TAKE UP THE MIDDLE BIGGEST NUMBER (12) AND PUT IT BACK AS IT WAS BEFORE THE DIGITS WERE ADDED TOGETHER AND YOUR ANSWER SHOULD BE 84