The number of triangular islands on a triangular grid

Cardinality of height function’s range in case of maximally many rectangular islands — computed by cuts

“…Pluhár [18] gave upper and lower bounds in higher dimensions. Horváth, Németh and Pluhár determined upper and lower bounds for the maximum number of triangular islands on a triangular grid in [9]. Some further interesting investigations and nice results on islands appear in [12,14,17].…”

“…Some further interesting investigations and nice results on islands appear in [12,14,17]. The number of square islands is a similar problem to the triangular case and it is treated in [7,13]. Some proving methods for the maximum number of islands are summarized in [1,16].…”

The possible number of islands on the sea

“…Several papers have been published on the subject since, investigating various extensions and generalizations (see G. Pluhár [8], E.K. Horváth, Z. Németh and G. Pluhár [4] and E.K. Horváth, G. Horváth, Z. Németh and Cs.…”

Elementary Approaches to the Teaching of the Combinatorial Problem of Rectangular Islands

“…It is noted that even if the obtained lower and upper bounds would have been different, the efforts for establishing those bounds are not in vain and there is no a priori guarantee that the exact solution has a nice closed form. In fact, if the number of triangular islands in a grid with size n is denoted by f(n), according to Horváth et al (2009), it is ''only'' known that n 2 þ3n 5 f ðnÞ 3n 2 þ9nþ2 14 .…”

“…Obviously this might seem to be somewhat random choice, because we may have also considered square islands on a rectangular or square grid. Moreover, instead of a rectangular grid we may also have considered islands on a triangular grid (Horváth et al 2009) or a torus (Barát et al submitted). Finally, we might have changed the neighbourhood relation (Barát et al submitted) or considered the corresponding counting problem in higher dimensions.…”