Eikooc Sushi play gamethe game play tree having n node number 1 n Recall tree having n node undirecte connect graph n1 edgesthey turn alternately move token tree Eikooc make place token node choice Sushi make follow Eikooc follow Sushi in turn player token node u u adjacent node v token currently u visit u oplus v leq minu v here x oplus y denote bitwise XOR operation integer x yBoth player play optimally the player unable losesthe follow example demonstrate rule game suppose Eikooc start game place token node 4 Sushi move token node 6 unvisited adjacent 4 it hold 6 oplus 4 2 leq min6 4 in turn Eikooc move token node 5 unvisited adjacent 6 it hold 5 oplus 6 3 leq min5 6 Sushi move play lose suppose Eikooc start game place token node 3 Sushi move token node 2 unvisited adjacent 3 it hold 3 oplus 2 1 leq min3 2 Eikooc token node 6 6 oplus 2 4 nleq min6 2 since Eikooc move play lose before game begin Eikooc decide sneakily relabel node tree favour formally relabele permutation p length n sequence n integer integer 1 n occur exactly p_i denote new number node iShe want maximize number node choose turn guarantee win help Eikooc find relabele help