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ABEL HAS 50, BILL HAS 20 AND CLARK HAS 30. ABEL ON HIS FIRST TURN OBVIOUSLY DOESN'T KNOW WHETHER HIS NUMBER IS 50 OR 10. SIMILARLY NEITHER BILL NOR CLARK CAN IMMEDIATELY FIGURE OUT THEIR NUMBERS. HOWEVER, ON HIS SECOND TURN ABEL CAN REASON: IF MINE IS A 10, THEN CLARK WOULD KNOW HIS NUMBER IS EITHER 10 OR 30. IF IT IS 10, BILL WOULD IMMEDIATELY KNOW HIS NUMBER IS 20. BUT HE DIDN'T KNOW. SO CLARK SHOULD KNOW HIS NUMBER IS 30. NOW SINCE CLARK DIDN'T KNOW, MY NUMBER MUST BE 50. WITH THIS KIND OF REASONING WE CAN ALSO RULE OUT ALL OTHER COMBINATIONS. SO [50, 20, 30] IS THE ONLY SOLUTION TO THIS PUZZLE 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 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 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