Power of three

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

Kalai's 3d conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope.

[5] 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.

[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.

However, the actual publication of the proof by Ronald Graham used a different number which is a power of two and much smaller.