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.”
“Envelopes? What envelopes?” you ask.
“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:
Advisor #1: Red Hat
Advisor #2: White Hat
Advisor #3: Red Hat
Advisor #4: White Hat
Advisor #5: Red Hat
Advisor #6: White Hat.
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?