| Dictionary: magic number |
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| Sci-Tech Encyclopedia: Magic numbers |
The number of neutrons or protons in nuclei which are required to fill major quantum shells. They occur at particle numbers 2, 8, 20, 50, and 82.
In atoms, the electrons that orbit the nucleus fill quantum electron shells at atomic numbers of 2 (helium), 10 (neon), 18 (argon), 36 (krypton), and 54 (xenon). These elements are chemically inert and difficult to ionize because the energies of orbits are grouped in bunches or shells with large gaps between them. In nuclei, an analogous behavior is found; quantum orbits completely filled with neutrons or protons result in extra stability. The neutrons and protons fill their quantum states independently, so that both full neutron and full proton shells can occur as magic nuclei. In a few cases, for example oxygen-16 (168O8) and calcium-40 (4020Ca20), doubly magic nuclei have full neutron and proton shells. Between the major shell gaps, smaller subshell gaps cause some extra stabilization and semimagic behavior is found at particle numbers 14, 28, 40, and 64. See also Atomic structure and spectra; Electron configuration.
In very heavy nuclei the Coulomb repulsion between the protons results in a different sequence of states for neutrons and protons and different major shell gaps. For neutrons the magic sequence continues at N = 126; the next shell gap is predicted at N = 184. For protons the next major shell gap is anticipated at Z = 114. The latter shell gaps lie beyond the heaviest nuclei known, but calculations indicate that the extra stability gained by producing nuclei with these particle numbers may result in an island of long-lived superheavy nuclei.
The closing of nuclear quantum shells has many observable consequences. The nuclei are more tightly bound than average, and the extra stability leads to anomalously high abundances of magic nuclei in nature. The full shells require unusually high energies to remove the least bound neutron or proton, and the probability of capturing extra particles is lower than expected. Furthermore, the full shells are spherically symmetric, and the nuclei have very small electric quadrupole moments. Many of these properties were known before the nuclear shell model was developed to account for quantum-level ordering and gaps between major shells. The different shell closures for atomic and nuclear systems reflect the differences between the Coulomb force that binds electrons to nuclei and the strong force that holds the nucleus together. An important component of the strong force in nuclei is the spin-orbit term, which makes the energy of a state strongly dependent on the relative orientation of spin and orbital angular momentum. See also
| Hacker Slang: magic number |
[Unix/C; common]
1. In source code, some non-obvious constant whose value is significant to the operation of a program and that is inserted inconspicuously in-line (hardcoded), rather than expanded in by a symbol set by a commented #define. Magic numbers in this sense are bad style.
2. A number that encodes critical information used in an algorithm in some opaque way. The classic examples of these are the numbers used in hash or CRC functions, or the coefficients in a linear congruential generator for pseudo-random numbers. This sense actually predates and was ancestral to the more common sense
3. Special data located at the beginning of a binary data file to indicate its type to a utility. Under Unix, the system and various applications programs (especially the linker) distinguish between types of executable file by looking for a magic number. Once upon a time, these magic numbers were PDP-11 branch instructions that skipped over header data to the start of executable code; 0407, for example, was octal for ‘branch 16 bytes relative’. Many other kinds of files now have magic numbers somewhere; some magic numbers are, in fact, strings, like the !<arch> at the beginning of a Unix archive file or the %! leading PostScript files. Nowadays only a wizard knows the spells to create magic numbers. How do you choose a fresh magic number of your own? Simple — you pick one at random. See? It's magic!
4. An input that leads to a computational boundary condition, where algorithm behavior becomes discontinuous. Numeric overflows (particularly with signed data types) and run-time errors (divide by zero, stack overflows) are indications of magic numbers. The Y2K scare was probably the most notorious magic number non-incident.
The magic number, on the other hand, is 7±2. See The magical number seven, plus or minus two: some limits on our capacity for processing information by George Miller, in the Psychological Review 63:81-97 (1956). This classic paper established the number of distinct items (such as numeric digits) that humans can hold in short-term memory. Among other things, this strongly influenced the interface design of the phone system.
| Measures and Units: magic number |
sub-atomic physics The number of electrons else of protons/neutrons characterizing very stable conditions, corresponding to filled shells in the relevant structure. Magic numbers include the atomic numbers of the noble gases (helium = 2, neon = 10, argon = 18, krypton = 36, xenon = 54, radon = 86) and, for nuclei, the values 2, 8, 20, 28, 50, 82, and 126, plus possibly higher values for artificial elements.
[Haxel O., Jensen J. H. D., Suess H. D. Phys. Rev. Vol. 75, 1766 (1949)]
| Wikipedia: Magic number (sports) |
In certain sports, a magic number is a number used to indicate how close a front-running team is to clinching a season title. It represents the total of additional wins by the front-running team or additional losses (or any combination thereof) by the rival team after which it is mathematically impossible for the rival team to capture the title in the remaining games. This assumes that each game results in a win or a loss, but not a tie. Teams other than the front-running team have what is called an elimination number (or "tragic number") (often abbreviated E#). This number represents the number of wins by the leading team or losses by the trailing team which will eliminate the trailing team. The elimination number for the second place team is exactly the magic number for the leading team.
The magic number is calculated as G + 1 − WA − LB, where
For example, in Major League Baseball there are 162 games in a season. Suppose the top of the division standings late in the season are as follows:
| Team | Wins | Losses |
| A | 96 | 58 |
| B | 93 | 62 |
Then the magic number for Team A to win the division is 162 + 1 − 96 − 62 = 5.
Any combination of wins by Team A and losses by Team B totaling to 5 makes it impossible for Team B to win the division title.
The "+1" in the formula serves the purpose of eliminating ties; without it, if the magic number were to decrease to zero and stay there, the two teams in question would wind up with identical records. If circumstances dictate that the front-running team would win the tiebreaker regardless of any future results, then the additional constant 1 can be eliminated. For example, the NBA uses complicated formulae for breaking ties, using several other statistics of merit besides overall win/loss record; however the first tiebreaker between two teams is their head-to-head record; if the frontrunning team has already clinched the better head-to-head record, then the +1 is unnecessary.
The magic number can also be calculated as WB + GRB - WA + 1, where
This second formula basically says: assume Team B wins every remaining game and then calculate how many games team A needs to win to surpass team B's maximum total by 1. Using the example above and with the same 162-game season, team B has 7 games remaining.
The magic number for Team A to win the division is still "5": 93 + 7 − 96 + 1 = 5.
Team B can win as many as 100 games. If Team A wins 101, Team B is eliminated. The magic number would decrease with a Team A win and would also decrease with a Team B loss, as its maximum win total would decrease by one.
In some sports, ties are broken by an additional one-game playoff(s) between the teams involved. When a team gets to the point where its magic number is 1, it is said to have "clinched a tie" for the division or the wild card. However, if they end the season tied with another team, and only one is eligible for the playoffs, the extra playoff game will erase that "clinching" for the team that loses the playoff game.
By convention, the magic number typically is used to describe the first place team only, relative to the teams it leads. However, the same mathematical formulas could be applied to any team, teams that are tied for the lead, as well as teams that trail. In these cases, a team that is not in first place will depend on the leading team to lose some games so that it may catch up, so the magic number will be larger than the number of games remaining. Ultimately, for teams that are no longer in contention, their magic number would be larger than their remaining games + the remaining games for the first place team - which would be impossible to overcome.
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The formula for the magic number is derived straightforwardly as follows. As before, at some particular point in the season let Team A have WA wins and LA losses. Suppose that at some later time, Team A has wA additional wins and lA additional losses, and define similarly WB, LB, wB, lB for Team B. The total number of wins that Team B needs to make up is thus given by (WA + wA) − (WB + wB). Team A clinches when this number exceeds the number of games Team B has remaining, since at that point Team B cannot make up the deficit even if Team A fails to win any more games. If there are a total of G games in the season, then the number of games remaining for Team B is given by G − (WB + wB + LB + lB). Thus the condition for Team A to clinch is that (WA + wA) − (WB + wB) = 1 + G − (WB + wB + LB + lB). Canceling the common terms, we obtain wA + lB = G + 1 − WA − LB, which establishes the magic number formula.
Sometimes a team can appear to have a mathematical chance to win even though they have actually been eliminated already, due to scheduling. In this Major League Baseball scenario, there are three games remaining in the season. Teams A, B and C are assumed to be eligible only for the division championship; another team with a better record in another division has already clinched the one available "wild card" spot:
| Team | Wins | Losses |
| A | 97 | 62 |
| B | 97 | 62 |
| C | 95 | 64 |
If Team C were to win all three remaining games, it would finish at 98-64, and if both Teams A and B were to lose their three remaining games, they would finish at 97-65, which would make Team C the division winner. However if Teams A and B are playing against each other in the final weekend (in a 3 game series), one of them will win at least two games and thereby clinch the division title with a record of either 100-62 or 99-63. The more direct consequence of this situation is that it is also not possible for Teams A and B to finish in a tie with each other, and Team C can't win the division
Another method can be used to determine the Elimination Number which uses only the Games Remaining and Games Behind Leader statistics, as follows: 
, where GR means Games Remaining, GBL means Games Behind Leader, L means Leader and T means Trailer. Note that this algorithm also is limited by the aforementioned Subtlties.
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
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 | Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more |
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| Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Read more |
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| Measures and Units. A Dictionary of Weights, Measures, and Units. Copyright © Donald Fenna 2002, 2004. All rights reserved. Read more |
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| Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Magic number (sports)". Read more |
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