Mutab, Neda, and Sogal

Today's puzzle comes from an old copy of Best of Creative Computing, Volume 1.  In 1976, Walter Koetke, of Lexington High School presented the following:

If you think you've seen this problem before, you may be correct. It's a really old problem in a new disguise.

The civilizations of three planets Neda, Mutab, and Sogal have agreed to begin war in the year 2431. Although these societies have not eliminated such irrational actions as war, they have at least formalized the process. There are, for instance, no guerrilla activities and wars are usually very brief and always decisive. Wars are fought with inter-planetary rockets each of which is powerful enough to completely destroy an entire planet. With such powerful weapons at their disposal, Neda, Mutab, ad Sogal have agreed to the following set of rules, for only in this way can they be assured of a single victor.

Rule 1: The fight will continue until only one civilization remains.
Rule 2: The rather primitive technique of drawing lots will be used to determine which planet may launch the first rocket, which the second, and which the third.
Rule 3: After launching rotation is established, rocket launching begins and continues in order until only one planet remains.

When contemplating the outcome of this war, the three civilizations have full knowledge of the background of their adversaries.

Mutab is clearly the technologically superior civilization. Once launched, their rockets always strike with perfect accuracy - thus disproving a modern theory that nothing is perfect. Before the war begins, both of the other civilizations are aware of the terrifying fact that if a Mutab rocket is fired at them, the probability of their being completely destroyed is 1.

Neda is the oldest civilization and long ago had the superior technology. However, the complacency of a self-centered, unchallenged mind has been eroding this superiority for many years. As a result, the technology of Neda has not advanced in over 40 years. If a Nedian rocket is fired at another planet, the probability of hitting that planet is 0.8, just as it was 40 years ago.

Sogal is by far the newest of the three civilizations. Being dedicated to producing its own technology on its own terms has resulted in a proud and purposeful civilization, but one that is technologically four or five hundred years behind its present adversaries. A missile launched by Sogal has only a 50-50 chance of reaching its intended target.

Your role in this future war is to determine each civilization's probability of winning.

Leap Babies

I'm sure you're aware that a year is defined as the number of days it takes for the Earth to revolve about the sun. That time is not evenly divisible into the number of hours it takes for the Earth to revolve on its axis, which is how we define the length of a day. What that all means is that instead of being exactly 365 days, a year is closer to 365¼ days.

In order to account for that extra ¼ day, we add an extra day to the calendar every four years and call that year Leap Year and the extra day is sometimes called Leap Day, which falls on February 29 in a Western calendar.

Now here's the puzzle for today: Assuming a regular birthrate, what percentage of the population celebrates their birthday on February 29?

Those Incredible Colored Socks

Professor Egghead woke up one morning and started dressing for a day in the office. He reached into his sock drawer and felt that there were four individual socks. He knew that two of those socks were black, one was white, and one was green. He grabbed two at random and put them on before leaving for work.

If one of the socks he was wearing was black, what is the probability that the other one was also black?

Happy Birthday, Professor!

Some time ago, Professor Egghead invited 5 of his colleagues from the university to help him celebrate his birthday. Each of the 5 guests were given a special party hat and Professor Egghead wore one too.

After the candles were blown out and everyone had eaten cake, Professor Egghead stood up to make an announcement. "Each of us is wearing a unique hat selected from my sizable collection of hats I've collected over the past several years while travelling to Australia, Canada, Germany, Holland, and Japan. Curiously, the six hats we're wearing tonight are either red, yellow, or blue, and I can see that there is at least one of each color. As a special party favour I've set aside a special gift for the first person who can figure out the color of the hat on their own head without taking it off or looking into a mirror."

Professor Egghead's friends all knew he had a knack for picking out especially nice gifts for party favours so, of course, they each wanted to be the first to figure out the color of their hat. They each looked around the room at the hats on the heads of the professor and each of the other guests but they couldn't figure out the solution.

Suddenly all the guests stood up at the same time and exclaimed "I know the color of the hat on my head!"

They were all right and Professor Egghead, with a twinkle in his eye, gave each of them a gift that he had picked out especially for them.


Why was it so hard for Professor Egghead's guests to figure out what color hat was on their own head and how did they suddenly come up with the right answer?

Knot or Not?

One day Professor Egghead was taking a walk in the park behind his house and spied a cord of some sort lying on the path. From the distance that he was from the rope he couldn't see if it was knotted or not.

As he continued to approach the cord he wondered about the probability of the cord being a knot.

In what orientations would the cord be a knot and what is the probability that the cord was knotted?