# Sunday Morning Puzzle #6

Sunday Morning Puzzle # 6 is a challenger. It is only recommended for superior logicians.

Great News! You’ve been nominated to be advisor to the king. It’s a great position meaning unbelievable wealth and prestige. Not to mention being a Royal Advisor is a great BABE MAGNET! Unfortunately, the king is one of THOSE kings. You know the kind. They select advisors based on the candidate’s ability to discern hat color. If you get this job one of your first pieces of advice to the king will be to discard this silly system.

The king’s advisory board is a seven member panel, one member of which has just resigned due to poor health. That’s the position you hope to fill. You know nothing about the rest of the advisors. They are cloaked in mystery. When you meet with the king he explains the rather complex rules to the 2001 Get A New Advisor Test:

“You will enter a room blindfolded and a hat will be placed on your head. The color of the hat will be either red or white. Then the blindfold will be removed. You will be seated at a round table in a plain room. Around you at the table, from left to right, will be Advisors #1 through #6–in that order. They will each also be wearing either a white or a red hat. You may not speak until your turn or you forfeit. Even when it is your turn you may only speak to name the color of your hat.”

“The advisors will each make one statement,” continues the king. “They will begin with Advisor #1 to your left, and continue around the table clockwise until it is your turn to speak. At your turn you may either (a) admit defeat and leave like a cowardly dog or (b) open one, and only one, of the envelopes in front of you on the table.”

“You shall see.” replies the king. “Once you open an envelope and read the contents you must name your hat color immediately. If you correctly name your hat color then you will be my next high-paid advisor. However, should you be wrong, you would be tossed into the fiery furnace. If you open your one envelope and still don’t know the color of your hat then you will also go into the furnace. If you cheat you will be beaten, quartered, and then tossed into the furnace–so don’t do that.”

“That’s a bit violent,” you say. “Anything else I need to know?”

“I guess I should tell you about the hats. First there are, of course, the traditional red hats and white hats. But there are two varieties of ‘special’ hats that may be worn. They are indistinguishable from the regular hats at first sight. One type is the Truth Hat. Whenever a true statement is spoken in the room, these hats change color–from either white to red, or from red to white. They will not change color if lies are told.”

“Then there are the Liar Hats.” continues the king. “These hats also change color, but only whenever a lie is told. Just like the Truth Hats, the Liar Hats go from either red to white, or from white to red.”

“NOW THIS IS IMPORTANT: ** There is an exception. A speaker’s hat can never change color because of the speaker’s statement. If someone lies while wearing a Liar Hat, then all other Liar Hats in the room change color, but not the speaker’s. The same goes for Truth Hats.** We won’t necessarily be using all four types of hats in our little test, but I thought you should be warned.”

“Let me get this straight,” you say. “There are four types of hats. The plain old traditional red hats and white hats that never change color may be used. But there are also two varieties of special hats that do change color. If I’m sitting around wearing any one of these four hats, nothing I might say could change the color of my own hat–regardless of it’s type. However, all other hats in the room that are either Truth Hats or Liar Hats would change color based on the veracity of my statements.”

“Right.” replies the king.

“And everyone else’s hats work the same way. So if, as an example, I am wearing a Liar Hat and Advisor #1 tells a lie, my hat and all other Liar Hats in the room will change color. But #1’s hat won’t change color–even if it is a Liar Hat.”

“Yep.” says the king.

“And if Advisor #1 makes a true statement then all Truth Hats in the room will change color–But #1’s hat would not–regardless of it’s type.”

“Right-O, you’ve got it.” replies the king in a jovial tone.

You are about to tell the king to take his hats and his test and stick them where the sun don’t shine when you notice a parade of lovely ladies-in-waiting stroll slowly by. Hmm… OK…

“Bring it on!” you tell the king.

The test begins. You are blindfolded and led into a room. A hat is placed on your head. Your blindfold is removed. Around you at the table, from left to right, sit the advisors. They stare at you with cold, blank eyes. The hats are as follows:

Then of course there is your own hat, which you cannot see. If any of these hats are “special” hats you cannot tell as of yet. You make a mental note that no advisor can see his own hat.

• Advisor #1 speaks: “I do not see more white hats than red hats.”

You look around to see if any of the hats change color. But, as far as you can tell, none do. Maybe this test won’t be so hard.

• Advisor #2 speaks: “I see four red hats.”

Immediately you see #3’s, #4’s and #6’s hats change color!

• Advisor #3 speaks: “I can see only one white hat.”

Now you see #2’s, #4’s, and #6’s hats change.

• Advisor #4 speaks: “Exactly two of us are blind.”

Oh now that’s just great. Grr. You do note, however, that no hats that you can see change color at this statement.

*Advisor #5 speaks: “I see three red hats, and three white hats.”

And here you see #2’s, #3’s, #4’s, and #6’s hats change color.

*Advisor #6 speaks: “I see exactly two white hats.”

At this statement you see the hats on #2, #3, and #4 change.

Now it is your turn. You think for a long time. Suddenly, you remember the envelopes! They lie in from of you on the otherwise bare table. Each envelope is titled. The first is called “Optometrist’s Report on Advisor #1.” The second envelope is “Optometrist’s Report on Advisor #2”. The third is such a report on Advisor #3. And so it goes down to the 6th envelope–which is of course the “Optometrist’s Report on Advisor #6”. Wow, you can open one of these envelopes to learn if a particular advisor is blind. But you can only read one report. Which one?

You think for a bit longer. Suddenly, you know what to do! You select a certain envelope, open it, and read the contents. Then you smile and announce the color of your hat.

Which envelope did you open, and why?

Wow. I’ve been looking at this for awhile. I’d have to say open up Envelope #1. If he is blind, then his statement could be true no matter what color hats anyone is wearing. This could also help you determine whether advisor 4 was telling the truth in saying “Two of us are blind.” Then you could determine whether hats #2, 3, 4 and 6 were truth hats or liar hats.
I may be way off right now. But that’s my guess, envelope #1.

This puzzle is extremely difficult as it takes several deductions to reach the final correct conclusion. Selecting the right envelope is the final step.

Please let the advisory board know what color your hat is— based on what information you find in the envelope. Or …face the FIERY FURNACE!! HAHAHAHAHAHAHA!!!

Hint: You must be certain to be able to determine your hat color regardless of which answer you get from the envelope. Which envelope gives you that certainty?

OK, my solution:

Choose envelope #1. If it says advisor #1 is blind, your hat is white; otherwise, your hat is red.

Here’s how I got to that:

There are six different hats you could be wearing. Either a plain red or white hat, or a red or white (at the start) Truth Hat, or a red or white (at the start) Liar Hat. Thus, there are six sequences of total visible hats to each advisor. Determining these sequences is academic and requires no real logic, except insofar as one must determine which sequence corresponds to a Truth and which to a Liar Hat.

It’s obvious that Hats 2, 3, 4, and 6 are either all Truth Hats, or all Liar Hats. If they are Truth Hats, then statements 1 and 4 are lies. Unfortunately, this throws into doubt the number of blind persons at the table. If we go with Occam’s Razor and assume no one’s blind, then Advisor 1 is telling a lie–but there’s no way to reconcile this in a single sequence with the other advisors’ allegedly true statements. Since we assume that the Mad Hatter King is interested in actually finding a new advisor and not merely furnace bait, and since it would be impossible to determine the number of blind advisors otherwise, we must take statement 4 as truth.

Thus: Advisors 2, 3, 5, and 6 lied, and Advisors 1 and 4 told the truth.

Taking this as correct, there are three sequences which can be reconciled by blindness in two advisors. If you’re wearing a plain hat, and Advisors 1 and 6 are blind, you have a white hat on. If you’re wearing a Truth Hat, and Advisors 4 and 5 are blind, you’re wearing red. If you’re wearing a Liar Hat, and Advisors 1 and 3 are blind, you’re wearing white. (All of these apply to the color of hat you’re wearing after all statements but yours have been made).

Note: Advisor 1 can be blind and give misleading information without having lied. He said, “I do not see more white hats than red hats.” This is true, even if he didn’t see anything.

So, there are three outcomes:

1 and 6 are blind: White
1 and 3 are blind: White (Liar)
4 and 5 are blind: Red (Truth)

Open envelope #1.
If it says Advisor 1 is blind, say, “I am wearing a white hat.”
If it says Advisor 1 has 20/20 vision, say, “I am currently wearing a red hat which will become white as soon as I finsih this statement.” Otherwise they can pull the hat off your head, show you that it’s white, and you’re furnace fodder.

So. How’d I do?

LL

Very nice puzzle Biotop. BTW, where will you be parking for the next few days? I feel the urge to slash your car tires.

Let’s get to the problem. First some observations and some terminology.

Let’s call the advisors A1, A2, …, and their respective statements S1, S2, … . We’ll be X.

Let’s say a hat that changes color on a lie is an L hat and a hat that changes color on a truth is a T hat.

T means true, F means false, R red, W white.

We know A2, A3, A4, A6 are wearing mutable hats. X might be wearing a mutable hat too so when I refer to X’s hat color I mean the color of X’s hat when the corresponding statement is made. Also a mutable hat toggles in color so after an even number of changes it is back to its initial color.

We have these 2 cases:

Case I : A2,A3,A4,A6 all wear L hats. S1=S4=T. S2=S3=S5=S6=F.

Case II : A2,A3,A4,A6 all wear T hats. S1=S4=F. S2=S3=S5=S6=T.
To determine the color of X’s hat, open the report on A5. Then we have these possibilities:

1. A5 is blind. Then S5=F since A5 sees nothing at all and we’re in Case I.

If A1 is sighted X=R to insure S1=T.

Say A1 is blind. Since S4=T A2 and A3 are sighted. S2=F implies X=W and S3=F implies X=R. So X is wearing a R hat of the L variety. Either way X initially wore a R hat.

1. A5 is sighted and speaks the truth. This is Case II. S5=T requires X=R and S6=T requires X=W. So X wears a W hat of the T variety.

2. A5 is sighted and lies. So S5=F and we’re in Case I again. S5=F requires X=W. But 2 false statements have occurred prior to S5. Whether X’s hat is mutable or not, X initially wore a W hat.

I suppose it is possible to determine whether X wears a mutable hat in 3)by considering the possible combinations of blind A’s but frankly I don’t care.

Congratulations to the new advisor to the king: jcgmoi. Your explanation is quite good. However, please don’t slash my tires. Such actions would not part of the approved code of conduct for king’s advisors.

Runner-up award to LazarusLong42. The benevolent king would never toss someone whose answer was so close to correct into the fiery furnace. The king is especially impressed with your detailed laying out of the puzzle, even though there was an error in the Red(Truth)explanation. Also of course remember that when the candidate announces the color of his hat, he needn’t worry about those words changing the hat’s color. No hat can change color as a result of its wearer’s statement. LL42 will be the first-runner-up. In the event that jcgmoi cannot fulfill the duties of the post( i.e. ladies-in-waiting), it will fall to the runner-up to do the job.