This page contains numbers appearing in combinatorics, which don’t fit on other lists.
List of numbers appearing in combinatorics[]
- In the Gregorian calendar, there are 217 combinations of day of the week and day in the month, Friday the 13th being the most (in)famous.
- Article 79 of the Basic Law for the Federal Republic of Germany requires constitutional amendments to be approved by an absolute two-thirds majority of the Bundestag along with a simple two-thirds majority of the Bundesrat. Article 51 of the same law gives each state at least three votes, a fourth vote for states with more than 2 million inhabitants, a fifth vote for states with more than 6 million inhabitants, and a sixth vote for states with more than 7 million inhabitants. There are currently four states with three votes, seven states with four votes, one state with five votes, and four states with six votes. A calculation reveals that of the 65,536 possible voting patterns, 7,228 lead to an absolute two-thirds majority.
- The number 7,825 is the smallest natural number n for which the set {1, 2, 3, … , n} cannot be written as a union of two disjoint sets, such that both of them contain no Pythagorean triples.[1]
- There are 17,152 solutions to the Ostomachion of Archimedes. However, this value is disputed, as it allows reflections of pieces and is just for square solutions, and not rectangles.[2]
- 3,628,800 is not only 10!, but is also the number of seconds in six weeks.
- The Kubo character has 11,007 possible mouth expressions and 4,429 possible brow expressions, and therefore 11,007×4,429 = 48,750,003 possible facial expressions.
- A 2011 paper by Coward and Lackenby[3] provides an upper bound on the number of Reidemeister moves required to pass between two knots with the same number of crossings. The upper bound given is \(2 \uparrow \uparrow (2^{163 \times 2^{14} \times n}) \approx 2 \uparrow \uparrow (10^{803,928n})\) (where n is the number of crossings), which they simplify to the larger bound \(2 \uparrow \uparrow (10^{1,000,000n})\). This is one of the largest numbers ever used in a mathematical proof.
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- 405 is the sum of all the numbers on a 9×9 Sudoku grid. It is equal to 9×T9, where Tn is the nth triangular number. This number is also called Ternary-dust mite.
- 6,670,903,752,021,072,936,960 is a combinatoric number equal to the number of possible 9×9 Sudoku grids.[4]
- 6,670,903,752,021,072,936,960 is roughly equal to \(f_2(66)\) in the fast-growing hierarchy.
- 109,110,688,415,571,316,480,344,899,355,894,085,582,848,000,000,000 is the product of all the numbers on a 9×9 Sudoku grid. It is equal to 19×29×39×49×59×69×79×89×99, the product of the first 9 9th powers.