174 (number)

174 (one hundred [and] seventy-four) is the natural number following 173 and preceding 175.

← 173 174 175 →
← 170 171 172 173 174 175 176 177 178 179 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred seventy-four
Ordinal	174th (one hundred seventy-fourth)
Factorization	2 × 3 × 29
Divisors	1, 2, 3, 6, 29, 58, 87, 174
Greek numeral	ΡΟΔ´
Roman numeral	CLXXIV
Binary	101011102
Ternary	201103
Senary	4506
Octal	2568
Duodecimal	12612
Hexadecimal	AE16

In mathematics

There are 174 7-crossing semi-meanders, ways of arranging a semi-infinite curve in the plane so that it crosses a straight line seven times.^[1] There are 174 invertible ${\displaystyle 3\times 3}$ (0,1)-matrices.^[2][3] There are also 174 combinatorially distinct ways of subdividing a topological cuboid into a mesh of tetrahedra, without adding extra vertices, although not all can be represented geometrically by flat-sided polyhedra.^[4]

The Mordell curve ${\displaystyle y^{2}=x^{3}-174}$ has rank three, and 174 is the smallest positive integer for which ${\displaystyle y^{2}=x^{3}-k}$ has this rank. The corresponding number for curves ${\displaystyle y^{2}=x^{3}+k}$ is 113.^[5][6]

In other fields

In English draughts or checkers, a common variation is the "three-move restriction", in which the first three moves by both players are chosen at random. There are 174 different choices for these moves, although some systems for choosing these moves further restrict them to a subset that is believed to lead to an even position.^[7]

See also

• The year AD 174 or 174 BC
• List of highways numbered 174
• All pages with titles containing 174

References

1. Sloane, N. J. A. (ed.). "Sequence A000682 (Semi-meanders: number of ways a semi-infinite directed curve can cross a straight line n times)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Sloane, N. J. A. (ed.). "Sequence A055165 (Number of invertible n X n matrices with entries equal to 0 or 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Živković, Miodrag (2006). "Classification of small (0,1) matrices". Linear Algebra and Its Applications. 414 (1): 310–346. arXiv:math/0511636. doi:10.1016/j.laa.2005.10.010. MR 2209249.
4. Pellerin, Jeanne; Verhetsel, Kilian; Remacle, Jean-François (December 2018). "There are 174 subdivisions of the hexahedron into tetrahedra". ACM Transactions on Graphics. 37 (6): 1–9. arXiv:1801.01288. doi:10.1145/3272127.3275037. S2CID 54136193.
5. Sloane, N. J. A. (ed.). "Sequence A031508 (Smallest k>0 such that the elliptic curve y^2 = x^3 - k has rank n, if k exists)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. Gebel, J.; Pethö, A.; Zimmer, H. G. (1998). "On Mordell's equation". Compositio Mathematica. 110 (3): 335–367. doi:10.1023/A:1000281602647. MR 1602064. S2CID 122592480. See table, p. 352.
7. Schaeffer, Jonathan (March 2005). "Solving checkers: first result". ICGA Journal. 28 (1): 32–36. doi:10.3233/icg-2005-28107.