L _ C K _ R
1 , 0 0 0 , 0 0 0
W _ L L
B _
_ P _ N .
T H _
N _ M B _ R
_ F
T _ M _ S
T H _ T
_ _ C H
L _ C K _ R
_ S
F L _ P P _ D
_ S
_ Q _ _ L
T _
T H _
N _ M B _ R
_ F
F _ C T _ R S
_ T
H _ S .
F _ R
_ X _ M P L _ ,
L _ C K _ R
1 2
H _ S
F _ C T _ R S
1 ,
2 ,
3 ,
4 ,
6 ,
_ N D
1 2 ,
_ N D
W _ L L
T H _ S
B _
F L _ P P _ D
6
T _ M _ S
( _ T
W _ L L
B _
F L _ P P _ D
W H _ N
Y _ _
F L _ P
_ V _ R Y
_ N _ ,
_ V _ R Y
2 N D ,
_ V _ R Y
3 R D ,
_ V _ R Y
4 T H ,
_ V _ R Y
6 T H ,
_ N D
_ V _ R Y
1 2 T H
L _ C K _ R ) .
I T
W _ L L
_ N D
_ P
C L _ S _ D ,
S _ N C _
F L _ P P _ N G
_ N
_ V _ N
N _ M B _ R
_ F
T _ M _ S
W _ L L
R _ T _ R N
_ T
T _
_ T S
S T _ R T _ N G
P _ S _ T _ _ N .
S _
_ F
_
L _ C K _ R
N _ M B _ R
H _ S
_ N
_ V _ N
N _ M B _ R
_ F
F _ C T _ R S ,
_ T
W _ L L
_ N D
_ P
C L _ S _ D .
I F
_ T
H _ S
_ N
_ D D
N _ M B _ R
_ F
F _ C T _ R S ,
_ T
W _ L L
_ N D
_ P
_ P _ N .
A S
_ T
T _ R N S
_ _ T ,
T H _
_ N L Y
T Y P _ S
_ F
N _ M B _ R S
T H _ T
H _ V _
_ N
_ D D
N _ M B _ R
_ F
F _ C T _ R S
_ R _
S Q _ _ R _ S .
T H _ S
_ S
B _ C _ _ S _
F _ C T _ R S
C _ M _
_ N
P _ _ R S ,
_ N D
F _ R
S Q _ _ R _ S ,
_ N _
_ F
T H _ S _
P _ _ R S
_ S
T H _
S Q _ _ R _
R _ _ T ,
W H _ C H
_ S
D _ P L _ C _ T _ D
_ N D
T H _ S
D _ _ S N ' T
C _ _ N T
T W _ C _
_ S
_
F _ C T _ R .
F _ R
_ X _ M P L _ ,
1 2 ' S
F _ C T _ R S
_ R _
1
X
1 2 ,
2
X
6 ,
_ N D
3
X
4
( 6
T _ T _ L
F _ C T _ R S ) .
O N
T H _
_ T H _ R
H _ N D ,
1 6 ' S
F _ C T _ R S
_ R _
1
X
1 6 ,
2
X
8 ,
_ N D
4
X
4
( 5
T _ T _ L
F _ C T _ R S ) .
S _
L _ C K _ R S
1 ,
4 ,
9 ,
1 6 ,
2 5 ,
_ T C . . .
W _ L L
_ L L
B _
_ P _ N .
S _ N C _
1 , 0 0 0 , 0 0 0
_ S
_
S Q _ _ R _
N _ M B _ R
( 1 0 0 0
X
1 0 0 0 ) ,
_ T
W _ L L
B _
_ P _ N
_ S
W _ L L Clue
THE MESSENGER HAS TO HAVE TRAVELED 2 KM. IT DOESN'T MATTER WHAT SPEED THEY WALKED AT. AT THE BEGINNING OF THE PUZZLE, THE LINE IS 1 KM LONG. THE GENERAL IS THEREFORE 1 KM AHEAD OF HIM. THE MESSENGER MUST THEREFORE TRAVEL MORE THAN 1 KM TO REACH THE GENERAL. SINCE THE LINE MOVES 1 KM FORWARD, THE END IS WHERE THE BEGINNING WAS. EVEN IF HE WALKED 1.5 KM TO THE GENERAL, HE ONLY HAS TO WALK 0.5 KM TO GET BACK TO THE END OF THE LINE. IT GOES FASTER GOING BACK, BECAUSE NOW THEY ARE COMING TOWARDS HIM, AND NOT GOING AWAY YOU WILL NEED TO OPEN A MINIMUM OF 2 BOXES. FIRST, OPEN THE BOX LABELLED APPLE; IF IT'S LABELLED CORRECTLY WE'RE DONE (THOUGH THIS IS PURELY BASED ON LUCK), OTHERWISE WE'LL FIND EITHER BANANAS, CARROTS, OR DATES. IN ANY OF THESE CASES, WE KNOW EITHER THAT THE BOXES LABELLED BANANAS, CARROTS, OR DATES MUST ALSO BE MISLABELED. IF THE FIRST OPENED BOX HAS: BANANAS INSIDE, THEN THE BOX LABELLED BANANA IS INCORRECT AND EITHER CARROTS OR DATES ARE CORRECT. CARROTS INSIDE, THEN THE BOX LABELLED CARROTS IS INCORRECT AND EITHER BANANA OR DATES ARE CORRECT DATES INSIDE, THEN THE BOX LABELLED DATES IS INCORRECT AND EITHER BANANA OR CARROTS ARE CORRECT. NO MATTER THE CIRCUMSTANCE, AFTER OPENING ONE BOX WE CAN IDENTIFY TWO REMAINING BOXES THAT MIGHT BE THE CORRECTLY LABELLED BOX. OPEN EITHER ONE OF THEM. IF IT'S THE CORRECTLY LABELLED BOX, WE'VE FOUND IT. OTHERWISE THE REMAINING UNOPENED BOX IS CORRECTLY LABELLED AND WE'VE FOUND IT. WE DON'T NEED TO OPEN IT TO DOUBLE CONFIRM LOCKER 1,000,000 WILL BE OPEN. THE NUMBER OF TIMES THAT EACH LOCKER IS FLIPPED IS EQUAL TO THE NUMBER OF FACTORS IT HAS. FOR EXAMPLE, LOCKER 12 HAS FACTORS 1, 2, 3, 4, 6, AND 12, AND WILL THUS BE FLIPPED 6 TIMES (IT WILL BE FLIPPED WHEN YOU FLIP EVERY ONE, EVERY 2ND, EVERY 3RD, EVERY 4TH, EVERY 6TH, AND EVERY 12TH LOCKER). IT WILL END UP CLOSED, SINCE FLIPPING AN EVEN NUMBER OF TIMES WILL RETURN IT TO ITS STARTING POSITION. SO IF A LOCKER NUMBER HAS AN EVEN NUMBER OF FACTORS, IT WILL END UP CLOSED. IF IT HAS AN ODD NUMBER OF FACTORS, IT WILL END UP OPEN. AS IT TURNS OUT, THE ONLY TYPES OF NUMBERS THAT HAVE AN ODD NUMBER OF FACTORS ARE SQUARES. THIS IS BECAUSE FACTORS COME IN PAIRS, AND FOR SQUARES, ONE OF THOSE PAIRS IS THE SQUARE ROOT, WHICH IS DUPLICATED AND THUS DOESN'T COUNT TWICE AS A FACTOR. FOR EXAMPLE, 12'S FACTORS ARE 1 X 12, 2 X 6, AND 3 X 4 (6 TOTAL FACTORS). ON THE OTHER HAND, 16'S FACTORS ARE 1 X 16, 2 X 8, AND 4 X 4 (5 TOTAL FACTORS). SO LOCKERS 1, 4, 9, 16, 25, ETC... WILL ALL BE OPEN. SINCE 1,000,000 IS A SQUARE NUMBER (1000 X 1000), IT WILL BE OPEN AS WELL THEIR HATS ARE ALL BLUE IN COLOR. THERE ARE 3 POSSIBLE HAT COLOR COMBINATIONS: [A] 1 BLUE, 2 WHITE [B] 2 BLUE, 1 WHITE [C] 3 BLUE THE COLOR COMBINATION OF 3 WHITE HATS IS NOT POSSIBLE SINCE THE KING HAS ALREADY SAID THAT AT LEAST ONE OF THE WISE MEN HAS A BLUE HAT. SO, LET'S START OUR ANALYSIS. WHAT IF THERE WERE ONE BLUE HAT AND TWO WHITE HATS? THEN THE WISE MAN WITH THE BLUE HAT WOULD HAVE SEEN TWO WHITE HATS AND IMMEDIATELY CALLED OUT THAT HIS OWN HAT WAS BLUE, SINCE HE KNEW THERE IS AT LEAST ONE BLUE HAT. THIS DIDN'T HAPPEN, AND THUS THE HAT COLOR COMBINATION [A] IS RULED OUT. NOW, THE WISE MEN KNEW THAT ONLY 2 HAT COLOR COMBINATIONS ARE POSSIBLE - COMBINATION [B] OR [C]. WHAT IF THERE WERE TWO BLUE HATS AND ONE WHITE HAT? THE MEN WITH THE BLUE HATS WILL SEE ONE WHITE HAT AND ONE BLUE HAT. THEY WILL CONCLUDE THAT COLOR COMBINATION [B] IS THE CASE AND WOULD CALL OUT BLUE AS THEIR HAT COLOR. THIS ALSO DID NOT HAPPEN, AND THUS THE HAT COLOR COMBINATION [B] IS ALSO RULED OUT. AFTER SOME TIME, WHEN NONE OF THE WISE MEN ARE ABLE TO IDENTIFY THE COLOR OF THEIR OWN HATS, COMBINATION [C] (OF 3 BLUE HATS) BECOMES THE ONLY POSSIBLE OPTION