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7 Clue
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 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 ADDRESS IS 1460 SUNSET BOULEVARD. YOU KNOW THAT THE HOUSE NUMBERS ARE EVEN AND CONSECUTIVE, SO THEY MUST BE APPROXIMATELY 1/6TH THE VALUE OF THE SUM 8790. IN FACT, THE NUMBER THAT IS 1/6TH THE TOTAL IS THE MEAN (AVERAGE) FOR ALL 6 HOUSES. THE AVERAGE NUMBER IS 1465 (8790 / 6). THERE MUST BE 3 HOUSE NUMBERS GREATER THAN THAT NUMBER, AND 3 HOUSE NUMBERS LESS THAN THAT NUMBER, ALL BEING EVEN AND CONSECUTIVE. THEREFORE, THE 6 HOUSE NUMBERS ARE 1460, 1462, 1464, 1466, 1468, 1470. THE LOWEST HOUSE NUMBER, AS PER THE QUESTION, IS THE ANSWER: 1460 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