Universalist

Two players take turns flipping a fair coin. The game ends when the same outcome occurs on three flips in a row. What is the probability that the player who goes first, wins?

An urn contains 3 red and 17 white balls. Balls are drawn without replacement. Find the probability that a red ball was drawn first, given that exactly one red ball was drawn on the 3rd, and 4th draws.

I have a column of positive numbers and want to take the natural log of each. I know how to do this for each individual number... but how do you get Excel to do it for the entire column automatically?

Please give me the sequence of commands or steps in detail. Thanks in advance.

Consider a square with a positive integer at each vertex subject to the following dynamics:

(1) Assign each edge the absolute value of the difference of the numbers on the adjacent vertices

(2) Erase the numbers at the vertices and replace them with the edge values

Show that if the process is repeated enough times, four zeros will be obtained.

The distance between Cairo and Damascus is 1000 miles. Your mission is to move a 10,000 kilogram load of grass from Cairo to Damascus using your camel, but you have two problems:

1) The camel won't budge unless you let it to continuously chew grass - it consumes 1 kilogram of grass per mile.

2) The camel's maximum load is 1000 kilograms.

What is the maximum amount of grass that you can get there?

1) A positive integer is written on a chalkboard. We repeatedly erase its unit digit and add five times that digit to what remains. Starting with 7^2007, can we ever end up at 2007^7?

#2) Determine the last two digits of 2^20000009 + 6^20000009 + 7^20000009.

the calculator's sliding case gets badly stuck. Is this a known problem to this model? Also, is there a remedy for it? Any input is appreciated.

I am trying to show that if A is a nxn Hermitian matrix (A* = A where A* is the adjoint aka conjugate transpose of A) then A is congruent to A^3 (the third power of A).

Congruent here means there is invertible S such that S*AS = A^3.

I also need a counter-example to show the same conclusion does not hold if A^3 is replaced with A^2.

This should have to do with the inertia of a matrix and Sylvester's law.

Solve the equation (ln x)^2 - 2.5(ln x)(ln(4x-5)) + (ln (4x-5))^2 = 0, where x and all expressions are real.

Find the number of positive factors of 36000000 which are NOT perfect squares.

You meet 3 witches. One always lies, one always tells the truth, and one sometimes lies and sometimes tells the truth. You get to ask 3 yes-no questions and must correctly identify which witch is which. The witches each know the truth telling policy of the other witches and each question must be addressed to only one witch who is the only one to answer. What questions should you ask and how do you interpret the replies?

An island contains cats, birds and snakes. Each day each cat eats a bird for breakfast, each snake has a cat for lunch and each bird has a snake for supper. There are no births or deaths except at these meals.

At a certain time on the n-th day of the month, a bird becomes the only living creature left on the island. What time of day did this happen (i.e. at what meal)? Find an expression (or write a program) to evaluate the number of birds before breakfast on the first day of the month, in terms of n.