81 (34) combinations of weights of 1 (30), 3 (31), 9 (32) and 27 (33) kg – each weight on the left pan, right pan or unused – allow integer weights from −40 to +40 kg to be balanced; the figure shows the positive values
In mathematics, a power of three is a number of the form 3n where n is an integer, that is, the result of exponentiation with number three as the base and integer n as the exponent.
In recreational mathematics and fractal geometry, inverse power-of-three lengths occur in the constructions leading to the Koch snowflake,[6]Cantor set,[7]Sierpinski carpet and Menger sponge, in the number of elements in the construction steps for a Sierpinski triangle, and in many formulas related to these sets. There are 3n possible states in an n-disk Tower of Hanoi puzzle or vertices in its associated Hanoi graph.[8] In a balance puzzle with w weighing steps, there are 3w possible outcomes (sequences where the scale tilts left or right or stays balanced); powers of three often arise in the solutions to these puzzles, and it has been suggested that (for similar reasons) the powers of three would make an ideal system of coins.[9]
^Tomita, Etsuji; Tanaka, Akira; Takahashi, Haruhisa (2006), "The worst-case time complexity for generating all maximal cliques and computational experiments", Theoretical Computer Science, 363 (1): 28–42, doi:10.1016/j.tcs.2006.06.015
^For the Brouwer–Haemers and Games graphs, see Bondarenko, Andriy V.; Radchenko, Danylo V. (2013), "On a family of strongly regular graphs with ", Journal of Combinatorial Theory, Series B, 103 (4): 521–531, arXiv:1201.0383, doi:10.1016/j.jctb.2013.05.005, MR 3071380. For the Berlekamp–van Lint–Seidel and Games graphs, see van Lint, J. H.; Brouwer, A. E. (1984), "Strongly regular graphs and partial geometries" (PDF), in Jackson, David M.; Vanstone, Scott A. (eds.), Enumeration and Design: Papers from the conference on combinatorics held at the University of Waterloo, Waterloo, Ont., June 14–July 2, 1982, London: Academic Press, pp. 85–122, MR 0782310
^Kalai, Gil (1989), "The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics, 5 (1): 389–391, doi:10.1007/BF01788696, MR 1554357, S2CID 8917264
^von Koch, Helge (1904), "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire", Arkiv för Matematik (in French), 1: 681–704, JFM 35.0387.02
^See, e.g., Mihăilă, Ioana (2004), "The rationals of the Cantor set", The College Mathematics Journal, 35 (4): 251–255, doi:10.2307/4146907, JSTOR 4146907, MR 2076132
^Gupta, Hansraj (1978), "Powers of 2 and sums of distinct powers of 3", Univerzitet u Beogradu Publikacije Elektrotehničkog Fakulteta, Serija Matematika i Fizika (602–633): 151–158 (1979), MR 0580438
^Gardner, Martin (November 1977), "In which joining sets of points leads into diverse (and diverting) paths", Scientific American, 237 (5): 18–28, Bibcode:1977SciAm.237e..18G, doi:10.1038/scientificamerican1177-18