david

A game is played as follows: Ten dice are thrown simultaneously and the dice that show the most common outcome are set aside [for example, three sixes are observed and those three dice are set aside]. The remaining dice are again thrown simultaneously and those that show the same number as the dice already set aside are also set aside. This continues until all the dice have shown the original number. How many total throws of the dice would this take, on average?

A random number p such that 0<p<1 is selected at random from a uniform distribution U[0,1]. A biased coin is then constructed such that the probability of heads on a single flip is p (thus 1-p for a tails).

This coin is flipped twice and the result is HH. If the coin is flipped a third time, what is the probability of a third head? More precisely, what is the fair value of a lottery ticket that pays one dollar if the third flip is a head?

A train accelerates from rest at a constant acceleration A. If a door of width L is open perpendicularly to the train as the train begins to accelerate, how long does the door take to close? (assume the door closes in the direction opposite to the train's motion).

The buses in a town are numbered 1 to N. You observe bus number 39. Estimate the number of buses in the town.

Consider the sequence a(n)=frac(1.5^n) , where frac(x) is the fractional part of x. Does this sequence get arbitrarily close to zero? In other words, what is lim(N->inf) of minimum (n=1 to N) (a(n)) ?

Evaluate Integral (0,inf) of (exp(-x)-exp(-2x))/x dx

When, where and by whom will a nuclear weapon next be used in anger?

A sphere rolls without slipping on the inside surface of a cylinder whose axis is vertical. Assuming no loss of energy due to friction, what is the motion in general? Does the sphere always fall downwards due to gravity, or is there a stable motion (presumably when the sphere is moving quickly) where it oscillates without moving downwards?

What are the three biggest technological improvements that have changed people's lives in the last 30 years?

Find ∫ [-inf,inf] exp(-x^2)*(erf(x/sqrt(2))^2 dx

where erf(z)=2/sqrt(pi)* ∫ [0,z] exp(-t^2)dt

I have an analytic answer but I can't prove it, not even close. Any ideas how to approach it?

I will assume familiarity with the rules. My question is how many cells need to be given in the puzzle, such that a unique solution is determined, and such that no single cell can be removed from the given data without causing uniqueness to be lost? Ie what is the size of the minimally sufficient set? Has any work been done on this?

How can you justify giving only a part of your wealth to charity? It seems that if you think charity is worthwhile, then you should give almost all your wealth to it. Either that or none. Anything in between is a cop out morally. What do people think?

Suppose in a hypothetical universe, the only matter present consists of two planets, one of which is spinning relative to the other, on an axis which joins the centers of the two planets. There is a man standing on the equator of each planet. Which man feels the centrifugal force? [By 'planet' here I am not implying a satellite, just a spherical body]

Is the answer different if these planets are in a deserted region of the present Universe?

Suppose a very fast light source revolves at 10,000 revolutions per second, and the light shines on a circular wall surrounding the source which has a circumference of 20 miles. Then the beam of light appearing on the wall moves at 200,000 miles per second, faster than the speed of light. Two questions (a) does the beam indeed move at 200,000 miles per second? (b) if so, how does this not violate relativity ?

I was calculating the infinite sum Sum(k=1 to inf) of 1/k^z for some complex numbers z and I noticed that whenever the series sums to zero, the real part of z is equal to 1/2. Can anyone confirm this is always true?