[bonanova]

The students of the senior class in the quaint town of Averitas all have exceptionally high IQ's.

For the record,

• Among the students with blond hair, boys outnumber girls 7 to 1.
  
• Among the students with red hair, girls outnumber boys 4 to 1.
  
• Among the girls, brunettes outnumber blonds 4 to 1 but are outnumbered 2 to 1 by redheads.
  
• Among the boys, as many are blond as are not blond.
  
• Among the brunette students, 10 are boys.
  

As a chaperone you are attending their graduation party, which has as the featured

event a magic show put on by the town Wizard. At the end of the show you hear him

address the students.

"Ya know guys, I've been doing these stupid magic shows year after year for longer

than I care to remember. And frankly they are beginning to bore me. So tonight,

[and with those words a sinister silence fell on the room] I'm going to perform a little

act that's not on the program. I'm going to cast a spell on you!

Until exactly one year from tonight, every one of you with red hair, when asked a question,

will by this spell be compelled to lie! Every brunette among you will by this spell be compelled

to tell the truth! And all of you with blond hair will by this spell be compelled to remain silent.

"But wait, there's more -- your spells will change!

"Whenever two of you with different spells meet, both of your spells will be changed

to the spell that neither of you had the moment you met. Take note of what I say:

your spells will be changed only if the spells you have at the moment you meet are

different. Meetings between students whose spells are the same at the moment they

meet will have no effect.

"Eventually your hair color won't identify your spell, and you will not know how

you will be compelled to respond to questions. Chaos will reign, and I, finally,

will have gotten a year of enjoyment from one of these idiotic magic shows!

[Evil laughter.]

"I am compassionate, however.

"I will return one year from tonight to this place and remove the spell. But, until

then, and beginning now" - as he raised his wand over the room - "the spell is in effect!"

You see the students begin to mill around the room wondering to each other how

they will get through the year, what with college entrance exams and the like.

But one boy walks alone to a chair and silently bows his head, apparently in a state

of shock. You walk over to him, notice his name tag, and address him:

"John, I understand this is a very serious predicament. But I am a puzzle solver,

so I don't really care how you're going to cope. What does interest me, tho, is this:

It occurs to me that if at some point in time you all come to have exactly the same

spell, no more changes will occur. Then things might start to settle down. If only

for a moment you all had the same curse then the surprise changes would end.

Do you think that's possible?" Might you all get the same curse at some point?

You see John think for a moment and shake his head slowly. "No," you hear him

say, "I don't think so."

Returning home, you relate your story to a fellow puzzle solver, who also thinks

for a moment. Then she says, "I think I know the color of John's hair!"

"Well," you say, "I've certainly given you enough information."

What color is John's hair?

Use the spoiler, Luke.

[Guest]

he is a brunette?

[Guest]

If, at any point, there are two hair colors which have the same total number of people, then it is trivial to get to the point where all have the same hair color.

The only way you can get to a point where two groups have the same total number is if any two groups start with numbers whose difference is a multiple of 3. This is because any switching will result in a -1,-1,+2 change, which means the differences will stay the same or change by 3.

This just leaves showing whether you start in a situation where there is a difference that's a multiple of 3.

[Guest]

He is a brunette. Reasoning being that there are 16 blonds, 18 brunettes and 20 red heads. It is impossible for these to meet in such a way as to only leave 1 type of spell, for the reason that has already been succinctly explained by Chuck.

[Guest]

I'm going to disagree with everybody (surprise!) and say John is a

carrot-top.

[Guest]

I'm going to disagree with everybody (surprise!) and say John is a

carrot-top.

Oops. It appears as though I killed off a couple of students by accident. Or maybe the magician made them disappear...

I'm guessing that's not allowed, making Chuck's solution correct..

I must be blond

Edited September 17, 2008 by d3k3

[Guest]

Blond. I'm guessing that since we were "in the room" when the spell was cast the spell is on us as well? It didn't say the spell only affected the students (correct me if I'm wrong). If that is the case then John would have to be a Blond since he didn't start talking until we met (which when we did) his spell was changed so that he could. Is it possible all of the ratios at the start of the problem were to throw us off?

[Guest]

Blond. I'm guessing that since we were "in the room" when the spell was cast the spell is on us as well? It didn't say the spell only affected the students (correct me if I'm wrong). If that is the case then John would have to be a Blond since he didn't start talking until we met (which when we did) his spell was changed so that he could. Is it possible all of the ratios at the start of the problem were to throw us off?

I guess it could go either way: "At the end of the show you hear him address the students " could mean the things he says are addressed only to the students, but not necessarily.

[bonanova]

Kudos to Chuck and Neida!