The magic hexagon at right is (up to symmetries) the only one which features the first integers (although larger magic hexagons exist which involve consecutive integers  not  starting with 1).  It's attributed to  Ernst von Haselberg  (1887)  but was rediscovered many times.

Wikipedia   |   Louis Hoelbling   |   Torsten Sillke

The way in which some of our physical units where designed means that some numbers have no reason whatsoever to be "round numbers".  For example, the fact that the speed of light (Einstein's constant) is very nearly  300 000 km/s  is a pure coincidence  (it's now equated de jure to 299792458 m/s to define the SI meter in term of the atomic second of time).  So is the fact that the diameter of the Earth is very nearly half a billion inches (the polar diameter is 500531678 inches, the equatorial diameter is 502215511 inches).

On the other hand, a few other "coincidences" were clearly either engineered or enhanced by metrologists...  Consider some "nearly correct" conversion factors:

• A typographer's point is 0.013837".
  That's about 72.2700007227 points to the inch. 
• An inch is 0.0254 m.
  That's about 39.37007874015748... inches to the meter.

In both cases, the very good accuracy of a rounded inverse conversion factor keeps alive a competing unit based on that.  The relevant numerical relations are:

  254 . 3937  =  999998
13837 . 7227  =  99999999

Pairs of decimal numbers which have roughly the same number of digits and are very nearly reciprocals of each other can be designed around the factorizations into primes of numbers close to  powers of ten.  For example:

Not all such pairs of factors are interesting, because...

In 1772, Leonhard Euler (1707-1783) found that  P(n)  =  n² + n + 41  is a prime number for any integer  n  from  0  to  39.  (It's divisible by 41 for n=40).

Note that   P(n-1)  =  P(-n)  =  n² - n + 41   (Legendre, 1798). Therefore, the above prime values of  P(n)  are duplicated when  n  goes down from -1 to -40.  So, there are 80 consecutive values of  n  (from -40 to +39)  which make  P(n)  prime  (each such prime number being obtained twice).

Thus, the polynomial   n² - (2q-1) n + (41+q²-q)   =   (n-q)² + (n-q) + 41   yields prime values for all integers from 0 to 39+q, provided that q is between 0 and 40.  In particular (for q=40) the polynomial   n² - 79n + 1601   yields only prime values as  n  goes from  0 to 79  (namely, 40 prime values appearing twice each)  as observed by  Hardy and Wright  in 1979.

Prime-Generating Polynomials  by Eric W. Weisstein

The challenge is to compute the integral   I = ò e^-x2 dx   which represents the area under a normal Gaussian curve.  The trick is to consider the square of this integral, which can be interpreted as a 2-dimensional integral which begs to be worked out in  circular  coordinates...  Unexpectedly, the result involves the constant p.

I²   =   ò e^-x2 dx   ò e^-y2 dy     =   òò e^-(x2+y2) dx dy    =   ò₀ 2pr e^-r2 dr
ò₀ 2pr e^-r2 dr   =   p ò₀ e^-u du   =   p

Therefore, the mysterious integral  I  is simply equal to  Öp

The most complicated is E8 (which may describe fundamental aspects of physical reality).  It describes the 248 ways to rotate an object with 57 dimensions.

The so-called Atlas Project culminated in an optimized computation about the representations of E8 which took 77 hours of supercomputer time to complete, on January 8, 2007.  The output was a square matrix of order 453060, having entries in a set of 1181642979 distinct polynomials totalizing 13721641221 integer coefficients with values up to 11808808...  all packed in  60 GB  of data.

The Scientific Promise of Perfect Symmetry:  A New-York Times article about E8, by Kenneth Chang
News about E8   by  John Baez  (2007-03-19)

The Fischer-Griess Monster Group  is also known as  Fischer's Monster,  or simply the  Monster Group.  It's (by far) the largest of the  sporadic groups.

It was predicted independently by Bernd Fischer and Robert L. Griess in 1973.  Calling it the  Friendly Giant,  Griess constructed it explicitely in 1981, as the automorphism group of a 196883-dimensional commutative nonassociative algebra over the rational numbers.

In 1978 (before that proof was completed) John McKay (1939-) spotted the appearance of the number 196884 in an expansion of the modular j-function (A000521) and subsequently wondered about some unexpected relation with the  Monster,  in a letter to John Thompson  (himself famous for the 1963  Feit-Thompson Theorem, which paved the road for a 20-year effort resulting in the final classification of finite groups).

Week 173  by  John Baez  (2001-11-25)