Finite Simple Group (of order two)

By their own admission, the Klein Four Group is the première a capella group in the world of higher mathematics. They are based at the Northwestern University mathematics department.


Much of Asia is in political turmoil, so I’ve had to make some difficult choices in my territories of Asia puzzle. Here, I’ll note some omissions and other relevant facts. Add your own comments below, but no flamewars please!

China: Oddly enough, even the very notion of ‘China’ is contentious, with two modern-day states vying for the title.

The last vestiges of ancient China were swept away in 1911 when the Qing dynasty was overthrown by a group of revolutionary factions. The most powerful of these was the Kuomintang, which went on to form the leadership of the autocratic Republic of China. They were overthrown by Mao Zedong’s Communist Party in 1950 and the People’s Republic of China was formed. The old Republic of China retreated to the island groups of Taiwan where they maintain authority. Until the lifting of martial law in 1987 they still insisted upon territorial rights to mainland China.

Israel/Palestine: Probably the most fought over strip of land in the history of this planet. It has been ruled by Canaanites, Israelites, Babylonians, Persians; the Greek, Roman, and Byzantine empires; by Sassanians, Omayyads, Crusaders, and Mamelukes. In ‘modern’ history, it was overseen by the Ottoman empire from 1517 to 1922. As part of war reparations after World War I, the area was given to Britain and became the British Mandate of Palestine, to be held in trust until such time as the region could govern itself. The problem was passed on to the UN and, finally, in 1948 it was decided that Palestine would be split into two territories, one Arab State and one Jewish State.

This decision was unpopular amongst the Arab population, who wanted the creation of a United State of Palestine. Soon after Israel declared its independence on 14th May 1948, around 30,000 troops from Syria, Iraq, Transjordan, Lebanon, and Egypt invaded the nascent state of Israel, beginning the 1948 Arab-Israeli War. At the end of the war, Israel had retained most of its territory, but had also gained half the land originally intended for the state of Palestine. The West Bank was annexed by Jordan, and the Gaza strip by Egypt.

Another flashpoint for the region occurred in 1956 during the Suez Crisis. The Suez Canal is a huge artificial canal west of the Sinai peninsula. It provides passage between Europe and Asia without requiring circumnavigation of Africa. In 1956 Egypt decided to nationalize the Suez Canal, effectively wresting control of the canal from European colonial powers. As a result of their decision, Britain and France invaded and captured the Sinai peninsula, supported by Israel. After a few months of Israeli occupation, the peninsula was given over to a United National Emergency Force (UNEF), charged with protecting the region from invasion.

The next major conflict occurred in 1967. Egypt expelled the UNEF from Sinai and began amassing troops on the border. Israel launched a pre-emptive strike on Egypt’s air force. Jordan weighed in by attacking western Jerusalem and Netanya. Syria shelled Galilee from the Golan Heights. By the end of the 1967 Arab-Israeli War (also called the Six-Day War), Israel had gained control of eastern Jerusalem, the Gaza Strip, the West Bank, the Sinai peninsula, and the Golan Heights.

Beginning in 1993 the Oslo Accords were an attempt to bring peace to the region, primarily by supporting Israeli withdrawal from parts of the Gaza Strip and the West Bank. In 2005, Israel unilaterally withdrew from the Gaza Strip. The region is now in Palestinian control, with power resting in the hands of Hamas, after the June 2007 Palestinian Civil War in which they overthrew the government of Fatah (essentially the PLO). The West Bank is now considered in international law to be a territory and not part of any state. The ultimate fate for the West Bank should follow the programme of the Oslo Accords.

I hope to write more about Asia, so please keep checking back!

read more | digg story

5 Pirates

Five ferocious pirates sacked Shanghai and escaped with 100 gold sovereigns. It’s now time to divide the booty between them. In decreasing order of seniority, they are: Able Ahab, Barnacle Bill, Crusty Chris, Dangerous Dave, and Eloquent Ed. Whatever they do, in dividing the loot, they must obey the Pirate’s Code. The Pirate’s Code: The most junior pirate begins by suggesting a distribution of the gold coins then all the pirates take a vote. If at least half of the pirates reject the offer then the junior pirate is executed and the bargaining continues with the next most junior pirate. Otherwise, the offer is accepted.

The Puzzle: Assume that the pirates are clever logicians, so that they can analyze this problem completely. Also assume that the pirates have the following priorities (in order): to protect their own lives, to maximize their profits, to kill their fellow pirates. What distribution of the gold sovereigns should Ed propose?

Hints and solutions available here.

Further thoughts: If you manage to solve this problem, think about what would happen if there were 6 pirates. To start with, we need to clarify one of the conditions. Rather than saying that the pirates try ‘to maximize their profits’ (a somewhat vague notion), we’ll insist that each pirate tries to maximize their minimum profit.

For example, Ahab may be faced with two choices. Following the first path, he may win 90 gold coins or 10 gold coins, depending on the choices of the other pirates. However, let’s suppose that following the second path he could win 50 gold coins or 11 gold coins. According to our principle, he’d choose the second path. The reason is that the minimum winnings are 10 coins and 11 coins respectively. So the second path maximizes his minimum profit.

Using this idea, you should be able to solve the 6 pirates problem, then solve the n-pirate problem for arbitrary n. Can you write a computer program that simulates the arbitrary problem?