CB

problem on arrangements of coins lying on the equilateral triangle latice Mamoru Watanabe ∗ This is a joint work with Kiyoshi Ando and Tomoki Nakamigawa. In this talk we will discuss a problem mentioned in a book [1]. Consider a equilateral triangle △ABC each of whose segment has length n and whose vertices are A, B and C. Mark each point on each peripheral segment S which has an integer distance from the endpoints of S, and add all straight segments passing through these points to be parallel to peripheral segments. Denote by Tn the figure given by this way and call it a equilateraltriangle latice. Let V (Tn) be the set of the vertices of Tn. Then Tn has just 1 2 (n+ 1)(n+ 2) vertices. There are many triangles whose vertices are in V (Tn). In [2], Nakamoto and Watanabe showed that the number of all triangles in Tn is given by ⌊18n(n+2)(2n+ 1)⌋. A subset H ⊆ V (Tn) is a destroyer if every three points of V (Tn) − H induce no triangle. We determined all the destroyers in V (Tn) (i ∈ {1, 2, 3, 4, 5,}) as given by the following theorem. Let Dn be the set of minimum destroyers and let ξ(n) be the size of a minimum destroyer. Let Bn be the arrangement a destroyer whose vertices are of V (Tn) − {1, 2, · · · , 0′} ∪ {0′′, 1′′, 2′′, · · · , (n − 2)′′, (n − 1)′′} (see Figure 2)