MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

118 Mathematical Recreations
Figure 4.22: Pool-Ball Triangle [133]
In a difference triangle, the consecutive numbers are arranged so that each
number below a pair of numbers is the positive difference between that pair.
He easily found two solutions for three balls and four solutions each for six and
ten balls. However, he was surprised to discover that, for all fifteen balls, there
is only the single solution shown in Figure 4.22, up to reflection. Incidentally,
it has been proved that no difference triangle can have six or more rows [295,
p. 7].
(a) (b)
Figure 4.23: Equilateral Triangular Billiards: (a) Triangular Pool Table. (b)
Unfolding Billiard Orbits. [198]
Recreation 20 (Equilateral Triangular Billiards [198]). In the “billiard
ball problem”, one seeks periodic motions of a billiard ball on a convex billiard
table, where the law of reflection at the boundary is that the angle of incidence
equals the angle of reflection [24, pp. 169-179]. Even for triangular pool tables,
the present state of our knowledge is very incomplete. For example, it is not
known if every obtuse triangle possesses a periodic orbit and, for a general
non-equilateral acute triangle, the only known periodic orbit is the Fagnano
orbit consisting of the pedal triangle (see Figure 2.35 right) [158]. However,
the equilateral triangular billiard table possesses infinitely many periodic orbits
[16].