bonanova

An urn contains b black balls and w white balls. Balls are removed singly and randomly until all the balls of one color are removed. What is the expected number of balls that remain in the urn?

We denote by factorial n! the product of the first n integers. 23! = 2585201ab38884976640000. Without performing multiplications, find the digits denoted here by a and b.

Does "when its 11:11:11 pm" refer to the actual time, or to the position of the hands?

I'm not sure of the meaning of "only Black's fifteen moves." Are we assuming anything at all about White's moves? Do White and Black move in turn as in an ordinary game?

Yes of course for the circle the locus of equidistant points is a circle with half the radius and thus 1/4 the area. The mistake of the first guess was to apply that reasoning to a polygon. m00li, agree it is an interesting problem, much deeper than first appearance.

I would agree that any pair of two people are either strangers or acquaintances. That is A cannot know B unless B also knows A, and A cannot be a stranger to B unless B is also a stranger to A. So ...

Since I inspired this, I guess I should solve it.

You may be right that not every set of four points will work as the OP asks. So can you construct a square on the points roughly as shown? Perhaps then derive conditions for the four points, that it can be done? This may be a richer problem that I originally thought.

You're right. I drew a ray from the center and took the midpoint. But that doesn't fit the question. I'll give it more thought.

Do we know the prior probability of having gingivitis?

Take any four points A, B, C, D in the plane, no three of which are collinear. They describe a unique quadrilateral, if we take the points as being its vertices. But they also describe a square, if we require only that the points lie on its sides. Using a compass and straightedge, construct a square such that the four points lie, one each, on its sides Edit: or the extensions of its sides. . . . . . . . . . . . . . . . Hint: The points are not special. Draw four similar points and do the construction on a sheet of paper if that helps. The answer would then be to describe the process.

Not really a proof.

Yeah. What he said. Prior ... Something. Bushindo explained Bayes to me once before, plasmid just did again, but it's still not something I could explain to my grandson.

Nice job m00li. It is the first of your two answers that I had in mind.

Determine the coordinates of six points on the plane with the following properties. No three points are collinear. Every pairwise distance is an integer. You may use sketchpad, compass, ruler, straight edge, whatever you think may be useful. The answer will be six pairs of coordinates: (xi , yi).

A while back I started a series of puzzles that seem difficult but have not-so-difficult answers. I can't find any of them, so I don't know how many exist. There must have been at least two, so I'll number this one 3. Have fun. A monotonic increasing function f (x) is cut above and below by horizontal lines that intersect it at f (x1) and f (x2). A vertical line is drawn through a point on the curve (black dot) between x1 and x2. The curve and the lines define the green and red areas shown on the figure. You are to find the point on the curve that minimizes the sum of these areas.

But they always move forward?

[spoiler=Looks like]Eric has one more toss than John so he has more heads OR more more tails but not both. That would require at least two tosses more than John. Since there is no bias, the two cases are equally likely, with probabilities of 1/2. The second question has a different answer, since we must rule out all cases where the number of heads are equal. I'm thinking.