Parcival

I just looked at Ian Stewart's book Math Hysteria. He covers this type of puzzle in Chapter 1.

I took Sepultallica's comments as a joke. Upon review, I see no unresolved question concerning the puzzle. Perhaps a version of the same puzzle with different chrome would help. Several hundred men are sitting in a large room. Each man has had either a red hat or a blue hat placed on his head. No man can see the hat on his own head, but can see the hat of every other man in the room. A (hatless) man enters the room carrying a bell. He explains that he will ring the bell every few minutes. When the bell is rung, every man who knows that he is wearing a blue hat should leave the room. He tells them at least one of them is wearing a blue hat. After 39 rings, no one has left the room. When the man rings the bell the 40th time, all the men wearing blue hats leave the room. How many men were wearing blue hats?

"the woman" was a typo. The priestess spoke all the women. Yes, all the adulterous husbands were shot. I was looking for the exact number. Each woman who shot her husband knew that he was an adulterer. Any other questions?

I searched the forum for any of the versions of this of which I am aware and found none. If I missed it, let me know. In a certain village, it is the moral duty of every woman to shoot her husband at midnight if she determines that he has been unfaithful. If a husband is unfaithful, every woman in the village, except his wife, knows it. On day, a priestess arrived in the village, told the woman that she knew of at least one adulterer in the village, and left. 39 nights passed without incident. Then at midnight on the 40th night, several women shot their husbands. How many adulterous husbands were shot that night?

A hint vis a vis finding y. Unfortunately, I don't see how knowing x, y, and z will help with the question as to whether the three tangents intersect. [hide]perimeter/2 = x + y +z.[/hide]

Not for all, surely. For an equilateral triangle with sides of length 2, y=1, but 2*area/(perimeter-2a) = 2*sqrt(3).

Right. Now, moving from algebra (back) to geometry, the (not necessarily right) triangle ABC has 3 G-circles, pairwise tangent at the points D, E, and F. Let d, e, and f be the lines tangent to the pairs of G-circles at the points D, E, and F, respectively. Do d, e, and f have a point in common?

Inspired by Grayfalcon's post. Consider a right triangle ABC, with a = 3, b = 4, and c = 5. Circles centered at A, B, and C of radii 3, 2, and 1, respectively, are pairwise externally tangent. Such circles have, to the best of my knowledge, no name. I shall refer to them as the triangle's G-circles. What are the radii of the G-circles of a right triangle of sides 5, 12, and 13? Of sides 20, 21, and 29?

"National Holiday Tree" is way too bland a name. Presumably, someone thinks that christmas trees are christian symbols. They were originally pagan, then christian, and now almost devoid of religious signifigance.