§6 annex: More about board sizes and shapes. This page elaborates on §6 of the home page.

§6X. This page presents a formula to count the number of hexagons on a board when they meet four criteria:

  1. the overall arrangement of the studs can be interpreted as a convex hexagon;
  2. no studs are missing;
  3. every stud has a tilable hexagon an each side;
  4. every tilable hexagon has exactly one adjacent stud.

Figure 6e1 is an irregular board not shown on the home page, while figure 6e2 is the same thing, but without the tilable hexagons:

figure 6e1 figure 6e2

Because the arrangement of studs can be interpreted as a hexagon (although with one very short edge) the quantities of studs on the six edges can be counted. Working clockwise from the upper left:

  studs
  labels qty
top s1, s2, s3, s4, s5, s6 a = 6
upper right s6, s13 b = 2
lower right s13, s20, s27 c = 3
bottom s27, s26, s25, s24, s23, s22, s21 d = 7
lower left s21 e = 1
upper left s21, s14, s7, s1 f = 4

This particular board was selected because every side is of a different length, reducing risk of ambiguity in the explanation.

It so happens that the following equations will automatically be satisfied:

a + b = d + e ⇒ 6 + 2 = 7 + 1
b + c = e + f ⇒ 2 + 3 = 1 + 4
c + d = f + a ⇒ 3 + 7 = 4 + 6

Now define the tri function:

tri (n) = n × (n − 1) ÷ 2

The number of studs s is:

s = tri (b + c + d − 1) − tri (b) − tri (d) − tri (f)
s = tri (2 + 3 + 7 − 1) − tri (2) − tri (7) − tri (4)
s = 55 − 1 − 21 − 6 = 27

The gross number of hexagons g is:

g = tri (2b + 2c + 2d − 1) − tri (2b) − tri (2d) − tri (2f)
g = tri (4 + 6 + 14 − 1) − tri (4) − tri (14) − tri (8)
g = 253 − 6 − 91 − 28 = 128

The number of tilable hexagons t is:

t = g − s
t = 128 − 27 = 101

Related topics from the present author:

§6Y. The next five images also appear on the home page.

The formula works for figure 6c1 below; but not 6c2, because its tiles 9 and 57 are not adjacent to any stud (criterion 4).

figure 6c1 figure 6c2

The formula fails for figure 6c3 below because of the hole (criterion 2), and fails for 6c4 because the overall shape of the board is neither convex nor a hexagon (criterion 1).

figure 6c3 figure 6c4

The formula fails for figure 7a because some studs (for example, s3) are not surrounded by six tilable hexagons (criterion 3).

figure 7a