Date: Thu, 31 Oct 1996 00:00:08

From: Alex Bogomolny

Hello,

Have you thought about the question you asked me? I came up with several properties of 30:

- 30 is the max number of edges that may be had by a regular convex polyhedron.
- 30 is the largest number such that all smaller numbers prime relative to it are actually prime.
- There are exactly 30 ways to color a cube with 6 colors.
- 12 is 30 in base 4 - the claim uses the first five digits.
- 30 is a
*primorial*number, viz., 30 = primorial(5) = 2*3*5. (primorial(p) is defined for prime p's and equals the product of all primes up to and including p.) - 30 is the smallest integer with three distinct prime divisors.
- There are only two Pythagorean triangles with perimeter equal their areas: 5-12-13 and 6-8-10. In the first case the number is 30, in the second 24.

Perhaps you can declare a competition in your math school. Somebody may actually come up with a novel property.

A useful book to look for unusual properties of numbers is

D.Wells, *The Penguin Dictionary of Curious and Interesting Numbers*, Penguin Books, 1986.

Best regards,

Alex