Power of three

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For other uses, see Power of Three.
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

exponent
.

Applications

The powers of three give the place values in the ternary numeral system.[1]

Graph theory

In

Berlekamp–van Lint–Seidel graph (243 vertices), and Games graph (729 vertices).[4]

Enumerative combinatorics

In enumerative combinatorics, there are 3n signed subsets of a set of n elements. In polyhedral combinatorics, the hypercube and all other Hanner polytopes have a number of faces (not counting the empty set as a face) that is a power of three. For example, a 2-cube, or square, has 4 vertices, 4 edges and 1 face, and 4 + 4 + 1 = 32. Kalai's 3d conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope.[5]

Inverse power of three lengths

In

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]

Perfect totient numbers

In number theory, all powers of three are perfect totient numbers.[10] The sums of distinct powers of three form a Stanley sequence, the lexicographically smallest sequence that does not contain an arithmetic progression of three elements.[11] A conjecture of Paul Erdős states that this sequence contains no powers of two other than 1, 4, and 256.[12]

Graham's number

Graham's number, an enormous number arising from a proof in Ramsey theory, is (in the version popularized by Martin Gardner) a power of three. However, the actual publication of the proof by Ronald Graham used a different number which is a power of two and much smaller.[13]

See also

References

  1. JSTOR 41185884
  2. S2CID 9855414
  3. doi:10.1016/j.tcs.2006.06.015
  4. ^ For the Brouwer–Haemers and Games graphs, see Bondarenko, Andriy V.; Radchenko, Danylo V. (2013), "On a family of strongly regular graphs with ",
    MR 0782310
  5. S2CID 8917264
  6. JFM 35.0387.02
  7. MR 2076132
  8. MR 3026271
  9. doi:10.1016/0165-1765(95)00691-8
  10. MR 2051959
  11. ^ Sloane, N. J. A. (ed.), "Sequence A005836", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
  12. MR 0580438
  13. doi:10.1038/scientificamerican1177-18
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