bonanova

Communication other than saying a color is banned. This solution violates the ban - it can be seen to fail if all that's communicated is red or black. If the loudness of the speaker's voice adds useful information, then it's a banned communication.

Doesn't the answer have to be a multiple of 5?

Here is an excellent source for finding sequences. Yours wasn't in it, however.

I would ask a different question: What digit actually comes after the 4?

Well if you're a liar, then the information about a ball in your hand is incorrect, and it would be hard to know what to ask.

Agree solution is the same - care to post it?

Kudos to Chuck and Neida!

Yup.

The solution with one extra point hasn't been found yet.

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.

Very nice. And thanks to Ben Law for the figure. Marinja correctly points out an error in the OP which restricts the solution to a single extra line. There is a simpler construction, but it instead adds a single extra point. And that point defines two extra lines. So to my knowledge a single extra line won't do it. My bad.

Images load ok - see this puzzle for instance. How are you doing it? You should click Browse > upload > Manage Current Attachments.

Here's an easy dissection problem in three dimensions. Cut a 12-inch cube into 4 parts. Re-assemble the parts into a 8"x8"x27" rectangular block. Have fun.

The solution to the first puzzle used two auxiliary lines. For example.

It might be. Can you show why?

Kudos!

Not sure I understand what you're asking.

Using a single straight-line cut?

It could be that this is exactly the same solution only explained differently - If so please accept my apologies It runs along the same line as octopuppy's solution, maybe easier to understand. Nice going.

Nope. Different angles.

What he said ... Hey Pallidus, welcome to the Den!

This one is a lot like the first, but it's simpler. Triangle ABC is isosceles. Find angle x. [edit - give the solution] Do not assume the figure is drawn to scale. Do not use the law of sines. Use simple geometric reasoning. Since it's easier, you may draw only one auxiliary line. Have fun.