模5不同余平面格点的形心问题研究

Study on the centroid of lattice points in a plane when lattice points are different after mod 5

对平面格点进行模5运算,建立了由25个不同剩余类格点排成5个行环、5个列环的环面格子网,称为模5环面,记为Z25.讨论了Z25上了格点之间、行之间、列之间、对角线之间的对称性,根据这些对称性得出了Z25上形心仍为格点的5个格点的分布情形.证明了当格点模5不同余时,任意9个格点中,必有5个格点其形心仍为格点,即公式n(5)=9成立.

Using modulo 5 （mod 5）operation of the lattice points in the plane, 25 lattice points are arranged into a ring surface lattice with 5 row rings and 5 column rings. The ring surface is called mod 5 ring surface, marked Z5^2. Symmetry between lattice points, between rows, between columns, and between diagonals are discussed, respectively. Based on these symmetries, it is pointed out that the distributing law of the 5 ! lattice points on Z5^2 whose centroid on is still a lattice point. It is proved that when lattice points have different remainders after mod 5 operation, in any 9 lattice points, there exist 5 lattice points whose centroid is still a lattice point, the formula n （5） = 9 found.