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P L _ C _ D Clue
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 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 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