joseboth[m_, mm_] := Block[{t, p, q, u, v, w}, w = mm - 1; t = Range[m]; p = t; q = t; Do[p = RotateLeft[p, w]; u = First[p]; p = Rest[p]; q = Drop[q, Position[q, u][[1]]]; If[Length[p] == 1, Break[],]; q = RotateRight[q, w]; v = Last[q]; q = Drop[q, -1]; p = Drop[p, Position[p, v][[1]]]; If[Length[q] == 1, Break[],], {n, 1, Ceiling[m/2]}]; p[[1]]];

The horizontal coordinate is for the number of numbers in the game, and the vertical coordinate is for the number that remains.

Please compare these two graphs．They have the same shape, and hence these graphs seem to have Fractal behavior (self-similarity). Note that = . This can be seen as the ratio of the self-similarity of the graph.

The authors could prove that the graph of in Example 2.2 has the self-similarity with the ratio of in [6]. The authors have not proved the existence of the self-similarity for . As to these problems the authors are going to present only graphs, and the graphs seem to have the self-similarity.

From Example 2.2, ..., Example 2.10 the authors can get the following prediction.