Thus named because it looks like arranging and rearranging huge numbers of draughtsmen in a giant parking lot.

Let's consider three types of numbers:
  • triangles: a(a+1)/2
  • squares: b2
  • centered hexagons: 3c(c+1) + 1
where a, b and c are natural numbers.

And two questions:

  • Which numbers are both triangular and square, triangular and hexagonal, square and hexagonal?
  • Which triangles divide into smaller triangles?
Solutions consist of one or more recurrent series of the type U(a,b,c)F, all with the same factor.

The first question
can be generalized to diophantine equations of the type p1a2+q1a + r1 = p2b2+q2b + r2, where pi, qi and ri are constants.
Here are the above mentioned examples:

Triangles through squares: a(a+1)/2 = b2
b:163520411896930...U(0,1,0)6, or the primus of 6.

Triangles through hexagons: a(a+1)/2 = 3c(c+1)+1

Squares through hexagons: b2 = 3c(c+1)+1
b:113181252135113489061...U(1,1,0)14, or the tertius of 14.

The second question
is about triangular division and thus about equations of the type a2+a = k(b2+b).
There are no solutions if k is a square. For non-square 'k', solutions approach the square root of 'k'.
Here are some examples:

a2+a = 2(b2+b)

a2+a = 3(b2+b)

a2+a = 5(b2+b)
a:0514992601785467432039838805749251505174...U(0,14,8)18 combined with U(-1,5,8)18.
b:0264411679820901432837512257114673134...U(0,6,8)18 combined with U(0,2,8)18.

a2+a = 6(b2+b)
a:0383584351836347982803444381968340955...U(0,8,4)10 combined with U(-1,3,4)10.
b:0131434143341142033801406133463139194...U(0,3,4)10 combined with U(0,1,4)10.