Cram is a mathematical game played on a sheet of graph paper. It is the impartial version of Domineering and the only difference in the rules is that each player may place their dominoes in either orientation, but it results in a very different game. It has been called by many names, including "plugg" by Geoffrey Mott-Smith, and "dots-and-pairs." Cram was popularized by Martin Gardner in Scientific American[1].
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Rules [link]
The game is played on a sheet of graph paper, with any set of designs traced out. It is most commonly played on rectangular board like a 6×6 square or a checkerboard, but it can also be played on an entirely irregular polygon or a cylindrical board.
Two players have a collection of dominoes which they place on the grid in turn. A player can place a domino either horizontally or vertically. Contrary to the related game of Domineering, the possible moves are the same for the two players, and Cram is then an impartial game.
As for all impartial games, there are two possible conventions for victory : in the normal game, the first player who cannot move loses, and on the contrary, in the misère version, the first player who cannot move wins.
Symmetry play [link]
The winning strategy for normal Cram is simple for even-by-even boards and even-by-odd boards. In the even-by-even case, the second player wins by symmetry play. This means that whichever move Player 1 makes, Player 2 has a corresponding symmetric move across the horizontal and vertical axes. In a sense, player 2 "mimics" the moves made by Player 1. If Player 2 follows this strategy, Player 2 will always make the last move, and thus win the game.
In the even-by-odd case, the first player wins by similar symmetry play. Player 1 places his first domino in the center two squares on the grid. Player 2 then makes his move, but Player 1 can play symmetrically thereafter, thus ensuring a win for Player 1.
It should be noted that symmetry play is a useless strategy in the misère version, because in that case it would only ensure the player that he loses.
Normal version [link]
Grundy value [link]
Since Cram is an impartial game, the Sprague–Grundy theorem indicates that in the normal version any Cram position is equivalent to a nim-heap of a given size, also called the Grundy value. Some values can be found in Winning Ways for your Mathematical Plays, in particular the 2 × n board, whose value is 0 if n is even and 1 if n is odd.
The symmetry strategy implies that even-by-even boards have a Grundy value of 0, but in the case of even-by-odd boards it only implies a Grundy value greater or equal to 1.
| n × m | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|
| 4 | 0 | 2 | 0 | 3 | 0 | 1 |
| 5 | - | 0 | 2 | 1 | 1 | ≥1 |
| 6 | - | - | 0 | >3 | 0 | ≥1 |
| 7 | - | - | - | ≥1 | ≥1 | ? |
Known values [link]
In 2009, Martin Schneider computed the grundy values up to the 3 × 9, 4 × 5 and 5 × 7 boards[2]. In 2010, Julien Lemoine and Simon Viennot applied to the game of Cram algorithms that were initially developed for the game of Sprouts[3]. It allowed them to compute the grundy-values up to the 3 × 18, 4 × 9 and 5 × 8 boards. They were also able to compute the outcome (but not the grundy-value) of the 5 × 9 and 7 × 7 boards[4].
The sequence of currently known Grundy values for 3 × n boards, from n=1 to n=18 is: 1, 1, 0, 1, 1, 4, 1, 3, 1, 2, 0, 1, 2, 3, 1, 4, 0, 1. It doesn't show any apparent pattern.
The table below details the known results for boards with both dimensions greater than 4. Since the value of an n × m board is the same as the value of a m × n board, we give only the upper part of the table.
Misère version [link]
Misère Grundy-value [link]
The misère Grundy-value of a game G is defined by Conway in On Numbers and Games as the unique number n such that G+n is a second player win in misère play[5]. Even if it looks very similar to the usual Grundy-value in normal play, it is not as much powerful. In particular, it is not possible to deduce the misère Grundy-value of a sum of games only from their respective misère grundy-values.
| n × m | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|
| 4 | 0 | 0 | 0 | 1 | 1 | 1 |
| 5 | - | 2 | 1 | 1 | ? | ? |
| 6 | - | - | 1 | ? | ? | ? |
Known values [link]
In 2009, Martin Schneider computed the misère grundy values up to the 3 × 9, 4 × 6, and 5 × 5 board[2]. In 2010, Julien Lemoine and Simon Viennot extended these results up to the 3 × 15, 4 × 9 and 5 × 7 boards, along with the value of the 6 × 6 board[4].
The sequence of currently known misère Grundy values for 3 × n boards, from n=1 to n=15 is: 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1. This sequence is conjectured to be periodic of period 3[4].
The table on the right details the known misère results for boards with both dimensions greater than 4.
References [link]
- Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (2003). Winning Ways for Your Mathematical Plays. A K Peters, Ltd..
- ^ Gardner, Martin (1974). "Mathematical Games: Cram, crosscram and quadraphage: new games having elusive winning strategies". Scientific American 230 (2): 106–108.
- ^ a b Das Spiel Juvavum, Martin Schneider, Master thesis, 2009
- ^ Julien, Lemoine; Simon, Viennot (2010). "Nimbers are inevitable". arXiv:1011.5841 [math.CO].
- ^ a b c Computation records of normal and misère Cram, Julien Lemoine and Simon Viennot web site
- ^ John H., Conway (2000). On Numbers and Games. A K Peters, Ltd..
https://wn.com/Cram_(game)
Cram (software)
Cram is an application for Apple's OS X and iOS developed by Patrick Chukwura and Ashli Norton of SimpleLeap Software.
The software is a flashcard application which allows users to prepare for various types of subject matter using flashcards and multiple choice tests. By entering the question and answer of the test in Cram, the application presents the information in test or flashcard format, which then allows the user to study the entered information at any time.
Apart from the core functionality of Cram, other features of the application include the use of images and sound that are integrated on the flashcard and practice tests as they study and test database that allows the user to download and share tests with other users.
Cram also provides functions to study from an iPhone with flashcards and multiple-choice tests.
Cram is available as shareware, which will block itself after creating five tests with five questions each.
See also
References
List of Middle-earth food and drink
The following list of Middle-earth food and drink includes all fictional items of food and drink featured in J. R. R. Tolkien's fantasy writings.
Foods
Cram
Appearing in The Hobbit and mentioned in The Lord of the Rings, cram is a biscuit-like food made by the Men of Esgaroth and Dale, which they share with the Dwarves of the Lonely Mountain. Very nutritious, it is used for sustenance on long journeys. It is not as appealing as and less tasty than the similar Elvish bread lembas; Tolkien describes it humorously as "more of a chewing exercise" than enjoyable to eat. Like lembas, it is probable that Tolkien modelled cram on hardtack, a biscuit that was used during long sea voyages and military campaigns as a primary foodstuff. This bread was little more than flour, water and salt which had been baked hard and would keep for months as long as it was kept dry.
Petty-dwarf roots
Petty-dwarf roots appear in some versions of the story of Túrin Turambar, as given in Unfinished Tales and The Children of Húrin. Túrin and the outlaws are sheltered by a petty-dwarf who gives them roots to cook. After being scrubbed and cooked, the roots are described as being fleshy and tasting like bread. The dwarf does not reveal what plant they are from, for fear of the plant's extinction, and refuses to teach the elves out of hatred. The "earthbread" is plainly a member of the potato family.
Pin
A pin is a device used for fastening objects or material together. Pins often have two components: a long body and sharp tip made of steel, or occasionally copper or brass, and a larger head often made of plastic. The sharpened body penetrates the material, while the larger head provides a driving surface. It is formed by drawing out a thin wire, sharpening the tip, and adding a head. Nails are related, but are typically larger. In machines and engineering, pins are commonly used as pivots, hinges, shafts, jigs, and fixtures to locate or hold parts.
Sewing and fashion pins
Curved sewing pins have been used for over four thousand years. Originally, they were fashioned out of iron and bone by the Sumerians and were used to hold clothes together. Later, these pins were also used to hold pages together by threading the needle through their top corner.
Many late pins were made of brass, a hard metal. Steel was used later, as it was much stronger, but there was no easy process to keep steel from rusting, so higher quality pins were plated with nickel, but the metal would start to break down and flake off in high humidity, allowing rust to form. Steel pins were not that inconvenient for homemaking uses as they were usually only used temporarily while sewing garments.
Łępin
Łępin [ˈwɛmpin] is a village in the administrative district of Gmina Stara Błotnica, within Białobrzegi County, Masovian Voivodeship, in east-central Poland. It lies approximately 15 kilometres (9 mi) south of Białobrzegi and 78 km (48 mi) south of Warsaw.
The village has a population of 148.
References
Coordinates: 51°30′58″N 20°58′1″E / 51.51611°N 20.96694°E
Pin (computer program)
Pin is a platform for creating analysis tools. A pin tool comprises instrumentation, analysis and callback routines. Instrumentation routines are called when code that has not yet been recompiled is about to be run, and enable the insertion of analysis routines. Analysis routines are called when the code associated with them is run. Callback routines are only called when specific conditions are met, or when a certain event has occurred. Pin provides an extensive application programming interface (API) for instrumentation at different abstraction levels, from one instruction to an entire binary module. It also supports callbacks for many events such as library loads, system calls, signals/exceptions and thread creation events.
Pin performs instrumentation by taking control of the program just after it loads into the memory. Then just-in-time recompiles (JIT) small sections of the binary code using pin just before it is ran. New instructions to perform analysis are added to the recompiled code. These new instructions come from the Pintool. A large array of optimization techniques are used to obtain the lowest possible running time and memory use overhead. As of June 2010, Pin's average base overhead is 30 percent (without running a pintool).
