This article relies largely or entirely on a single source .(January 2013) |

The **pirate game** is a simple mathematical game. It is a multi-player version of the ultimatum game.

There are five rational pirates (in strict order of seniority A, B, C, D and E) who found 100 gold coins. They must decide how to distribute them.

The pirate world's rules of distribution say that the most senior pirate first proposes a plan of distribution. The pirates, including the proposer, then vote on whether to accept this distribution. If the majority accepts the plan, the coins are dispersed and the game ends. In case of a tie vote, the proposer has the casting vote. If the majority rejects the plan, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again. The process repeats until a plan is accepted or if there is one pirate left.^{ [1] }

Pirates base their decisions on four factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins he receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.^{ [2] } And finally, the pirates do not trust each other, and will neither make nor honor any promises between pirates apart from a proposed distribution plan that gives a whole number of gold coins to each pirate.

To increase the chance of his plan being accepted, one might expect that Pirate A will have to offer the other pirates most of the gold. However, this is far from the theoretical result. When each of the pirates votes, they won't just be thinking about the current proposal, but also other outcomes down the line. In addition, the order of seniority is known in advance so each of them can accurately predict how the others might vote in any scenario. This becomes apparent if we work backwards.

The final possible scenario would have all the pirates except D and E thrown overboard. Since D is senior to E, he has the casting vote; so, D would propose to keep 100 for himself and 0 for E.

If there are three left (C, D and E), C knows that D will offer E 0 in the next round; therefore, C has to offer E one coin in this round to win E's vote. Therefore, when only three are left the allocation is C:99, D:0, E:1.

If B, C, D and E remain, B can offer 1 to D; because B has the casting vote, only D's vote is required. Thus, B proposes B:99, C:0, D:1, E:0.

(In the previous round, one might consider proposing B:99, C:0, D:0, E:1, as E knows it won't be possible to get more coins, if any, if E throws B overboard. But, as each pirate is eager to throw the others overboard, E would prefer to kill B, to get the same amount of gold from C.)

With this knowledge, A can count on C and E's support for the following allocation, which is the final solution:

- A: 98 coins
- B: 0 coins
- C: 1 coin
- D: 0 coins
- E: 1 coin
^{ [2] }

(Note: A:98, B:0, C:0, D:1, E:1 or other variants are not good enough, as D would rather throw A overboard to get the same amount of gold from B.)

The solution follows the same general pattern for other numbers of pirates and/or coins. However, the game changes in character when it is extended beyond there being twice as many pirates as there are coins. Ian Stewart wrote about Steve Omohundro's extension to an arbitrary number of pirates in the May 1999 edition of Scientific American and described the rather intricate pattern that emerges in the solution.^{ [2] }

Supposing there are just 100 gold pieces, then:

- Pirate #201 as captain can stay alive only by offering all the gold one each to the lowest
*odd*-numbered pirates, keeping none. - Pirate #202 as captain can stay alive only by taking no gold and offering one gold each to 100 pirates who would not receive a gold coin from #201. Therefore, there are 101 possible recipients of these one gold coin bribes being the 100
*even*-numbered pirates up to 200 and number #201. Since there are no constraints as to*which*100 of these 101 he will choose, any choice is equally good and he can be thought of as choosing at random. This is how chance begins to enter the considerations for higher-numbered pirates. - Pirate #203 as captain will not have enough gold available to bribe a majority, and so will die.
- Pirate #204 as captain has #203's vote secured without bribes: #203 will only survive if #204 also survives. So #204 can remain safe by reaching 102 votes by bribing 100 pirates with one gold coin each. This seems most likely to work by bribing
*odd*-numbered pirates optionally including #202, who will get nothing from #203. However, it may also be possible to bribe others instead as they only have a 100/101 chance of being offered a gold coin by pirate #202. - With 205 pirates, all pirates bar #205 prefer to kill #205 unless given gold, so #205 is doomed as captain.
- Similarly with 206 or 207 pirates, only votes of #205 to #206/7 are secured without gold which is insufficient votes, so #206 and #207 are also doomed.
- For 208 pirates, the votes of self-preservation from #205, #206, and #207 without any gold are enough to allow #208 to reach 104 votes and survive.

In general, if G is the number of gold pieces and N (> 2G) is the number of pirates, then

- All pirates whose number is less than or equal to 2G + M will survive, where M is the highest power of 2 that does not exceed N – 2G.
- Any pirates whose number exceeds 2G + M will die.
- Any pirate whose number is greater than 2G + M/2 will receive no gold.
- There is no unique solution as to who gets one gold coin and who does not if the number of pirates is 2G+2 or greater. A simple solution dishes out one gold to the
*odd*or*even*pirates up to 2G depending whether M is an even or odd power of 2.

Another way to see this is to realize that every M^{th} pirate will have the vote of all the pirates from M/2 + 1 to M out of self preservation since their survival is secured only with the survival of the Mth pirate. Because the highest ranking pirate can break the tie, the captain only needs the votes of half of the pirates over 2G, which only happens each time (2G + a Power of 2) is reached. For instance, with 100 gold pieces and 500 pirates, pirates #500 through #457 die, and then #456 survives (as 456 = 200 + 2^{8}) as he has the 128 guaranteed self-preservation votes of pirates #329 through #456, plus 100 votes from the pirates he bribes, making up the 228 votes that he needs. The numbers of pirates past #200 who can guarantee their survival as captain with 100 gold pieces are #201, #202, #204, #208, #216, #232, #264, #328, #456, #712, etc.: they are separated by longer and longer strings of pirates who are doomed no matter what division they propose.

- ↑ Bruce Talbot Coram (1998). Robert E. Goodin (ed.).
*The Theory of Institutional Design*(Paperback ed.). Cambridge University Press. pp. 99–100. ISBN 978-0-521-63643-8. - 1 2 3 Stewart, Ian (May 1999), "A Puzzle for Pirates" (PDF),
*Scientific American*, pp. 98–99

The **wave equation** is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

In probability theory, a **probability space** or a **probability triple** is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same.

In recreational mathematics and combinatorial design, a **magic square** is a square grid filled with distinct positive integers in the range such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the *magic constant* or *magic sum* of the magic square. A square grid with n cells on each side is said to have *order n*.

The **Universal Product Code** (**UPC**) is a barcode symbology that is widely used in the United States, Canada, Europe, Australia, New Zealand, and other countries for tracking trade items in stores.

A **penny** is a coin or a unit of currency in various countries. Borrowed from the Carolingian denarius, it is usually the smallest denomination within a currency system. Presently, it is the formal name of the British penny (abbr. **p**) and the informal name of the American one cent coin (abbr. **¢**) as well as the informal Irish designation of the 1 cent euro coin (abbr. **c**). It is the informal name of the cent unit of account in Canada, although one cent coins are no longer minted there. The name is also used in reference to various historical currencies also derived from the Carolingian system, such as the French denier and the German pfennig. It may also be informally used to refer to any similar smallest-denomination coin, such as the euro cent or Chinese fen.

In mathematics and physics, a **brachistochrone curve**, or curve of fastest descent, is the one lying on the plane between a point *A* and a lower point *B*, where *B* is not directly below *A*, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.

The **guinea** was a coin of approximately one quarter ounce of gold that was minted in Great Britain between 1663 and 1814. The name came from the Guinea region in West Africa, where much of the gold used to make the coins originated. It was the first English machine-struck gold coin, originally worth one pound sterling, equal to twenty shillings, but rises in the price of gold relative to silver caused the value of the guinea to increase, at times to as high as thirty shillings. From 1717 to 1816, its value was officially fixed at twenty-one shillings.

The **Paillier crypto system**, invented by and named after Pascal Paillier in 1999, is a probabilistic asymmetric algorithm for public key cryptography. The problem of computing *n*-th residue classes is believed to be computationally difficult. The decisional composite residuosity assumption is the intractability hypothesis upon which this cryptosystem is based.

In mathematics, a **closed-form expression** is a mathematical expression expressed using a finite number of standard operations. It may contain constants, variables, certain "well-known" operations, and functions, but usually no limit, differentiation or integration. The set of operations and functions admitted in a closed-form expression may vary with author and context.

The **Schulze method** is an electoral system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners. The Schulze method is also known as **Schwartz Sequential dropping** (**SSD**), **cloneproof Schwartz sequential dropping** (**CSSD**), the **beatpath method**, **beatpath winner**, **path voting**, and **path winner**.

**Circumscription** is a non-monotonic logic created by John McCarthy to formalize the common sense assumption that things are as expected unless otherwise specified. Circumscription was later used by McCarthy in an attempt to solve the frame problem. To implement circumscription in its initial formulation, McCarthy augmented first-order logic to allow the minimization of the extension of some predicates, where the extension of a predicate is the set of tuples of values the predicate is true on. This minimization is similar to the closed-world assumption that what is not known to be true is false.

The * Black Pearl* is a fictional ship in the

The **lifting scheme** is a technique for both designing wavelets and performing the discrete wavelet transform (DWT). In an implementation, it is often worthwhile to merge these steps and design the wavelet filters *while* performing the wavelet transform. This is then called the second-generation wavelet transform. The technique was introduced by Wim Sweldens.

**Phenylacetylene** is an alkyne hydrocarbon containing a phenyl group. It exists as a colorless, viscous liquid. In research, it is sometimes used as an analog for acetylene; being a liquid, it is easier to handle than acetylene gas.

**Bertrand's box paradox** is a paradox of elementary probability theory, first posed by Joseph Bertrand in his 1889 work *Calcul des probabilités*.

* Pirate Master* was a CBS reality television show created by Mark Burnett which replaced the previous Mark Burnett show on CBS, Rock Star. It followed sixteen modern-day pirates on their quest for gold, which totaled US$1,000,000. The show was hosted by Cameron Daddo, and took place in the Caribbean island nation of Dominica. The show premiered on Thursday, May 31, 2007. The show also aired on CTV in Canada, Sky3 in the UK, premiered on June 21, 2007 on Network Ten in Australia, and premiered on July 4, 2007 on AXN Asia. On July 10 in the US, the show moved to Tuesdays at 10 p.m. (ET). In its Tuesday run, it would follow the 9 p.m. (ET) broadcast of

The Mathieu equation is a linear second-order differential equation with periodic coefficients. The French mathematician, E. Léonard Mathieu, first introduced this family of differential equations, nowadays termed Mathieu equations, in his “Memoir on vibrations of an elliptic membrane” in 1868. "Mathieu functions are applicable to a wide variety of physical phenomena, e.g., diffraction, amplitude distortion, inverted pendulum, stability of a floating body, radio frequency quadrupole, and vibration in a medium with modulated density"

**The Ouse Valley Railway** was to have been part of the London, Brighton & South Coast Railway (LBSCR). It was authorised by an Act of Parliament and construction of the 20 miles (32 km) long line was begun, but not completed. It never opened to traffic.

A **balance puzzle** or **weighing puzzle** is a logic puzzle about balancing items—often coins—to determine which holds a different value, by using balance scales a limited number of times. These differ from puzzles that assign weights to items, in that only the relative mass of these items is relevant.

**Board puzzles with algebra of binary variables** ask players to locate the hidden objects based on a set of clue cells and their neighbors marked as variables (unknowns). A variable with value of 1 corresponds to a cell with an object. Contrary, a variable with value of 0 corresponds to an empty cell—no hidden object.

- Robert E. Goodin, ed. (1998). "Chapter 3: Second best theories".
*The Theory of Institutional Design*. Cambridge University Press. pp. 90–102. ISBN 978-0-521-63643-8.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.