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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 THE SECRET CODE IS 7,4,6,5,8. LET THE NUMBERS = A, B, C, D, E BASED ON 1ST CLUE: E + C = 14 BASED ON 2ND CLUE: A = 2B - 1 BASED ON 3RD CLUE: D = B + 1 BASED ON 4TH CLUE: B + C = 10 BASED ON 5TH CLUE: A + B + C + D + E = 30 ALL NUMBERS CAN BE REPRESENTED BY B EXCEPT FOR E SO WE NEED TO REWRITE EQUATION 1. SUBSTITUTE C = 10 - B FROM EQUATION 4 INTO EQUATION 1. E + (10 - B) = 14 E = B + 4 SUBSTITUTE A,C,D,E IN EQUATION 5 (2B - 1) + B + (10 - B) + (B + 1) + (B + 4) = 30 SOLVING FOR B GIVES 4. USE THE OTHER EQUATIONS TO FIND THE VALUES OF A,C,D AND E THE CODE IS 042. BASED ON CLUE #1 AND #2, WE CAN SAY THAT 6 IS NOT THE CORRECT NUMBER. NOW WE KNOW THAT THE UNLOCK CODE DOES NOT CONTAIN 6 AND BASED ON CLUE #3, WE KNOW THAT 2 AND 0 ARE CORRECT NUMBERS BUT IN THE WRONG POSITION. FROM CLUE #1, 2'S POSITION SHOULD BE ON THE RIGHT. THE CODE IS ??2. BASED ON CLUE #5, 0'S POSITION SHOULD BE ON THE LEFT. THE CODE IS 0?2. WE HAVE TO FIND THE MISSING NUMBER IN THE MIDDLE POSITION. BASED ON CLUE #2, THE MIDDLE NUMBER HAS TO BE 4. IT CANNOT BE 1, AS THE STATEMENT SAYS THAT THE NUMBER IS WRONGLY PLACED THE FOUR SOLUTIONS ARE 2-3, 3-4, 9-8 AND 8-7. THE TRICK IS TO UNDERSTAND THAT A PERSON IS ABLE TO DETERMINE THE UNKNOWN CONSECUTIVE NUMBER CORRECTLY ONLY WHEN THERE IS ONLY 1 POSSIBLE CHOICE. AFTER THE 1ST STATEMENT, WHEN A SAYS "I DO NOT KNOW YOUR NUMBER", IT IS CLEAR THAT THE NUMBER KNOWN TO HIM IS NEITHER 1 NOR 10, OTHERWISE HE WOULD HAVE KNOWN B'S NUMBER. SO, NOW, B KNOWS THAT A KNOWS SOME NUMBER OTHER THAN 1 OR 10. IF THE NUMBERS KNOWN TO B WERE 2 OR 9, HE COULD HAVE IMMEDIATELY DEDUCED THE FACT THAT THE NUMBER KNOWN TO A IS 3 OR 8. BUT SINCE, HE SAYS "NEITHER DO I KNOW YOUR NUMBER", IT MEANS THAT B'S NUMBER IS NOT 1, 2, 9, OR 10. HOWEVER, THIS DOES NOT MEAN THAT THE NUMBERS KNOWN TO A CAN'T BE 2 OR 3. FROM THE 3RD STATEMENT, A NOW KNOWS THE NUMBER, THEREFORE A'S NUMBER MUST BE 2, 3, 8 OR 9. SO, IF A'S NUMBER IS 2, THEN HE CAN BE SURE THAT B'S NUMBER IS 3, IF A'S NUMBER IS 3, THEN HE CAN BE SURE THAT B'S NUMBER IS 4, IF A'S NUMBER IS 9, THEN HE CAN BE SURE THAT B'S NUMBER IS 8, IF A'S NUMBER IS 8, THEN HE CAN BE SURE THAT B'S NUMBER IS 7