方阵填数 行列式为零

hbghlyj

甲乙二人在空方阵上交替地填数.甲先填.每人每次只能在空的位置填写一个实数.方阵被填满后游戏结束.如果最后方阵的行列式不为零,判甲胜.否则判乙胜.问甲乙二人谁有必胜的策略?
https://math.stackexchange.com/q ... ling-an-arrayputnam
Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
若甲在第一行填数,则乙在第二行的同一列的元素填相同的数.
若甲在第二行填数,则乙在第一行的同一列的元素填相同的数.
若甲在其他行填数,则乙在其他行任意找一个元素填任意的数.
最终第一行和第二行必完全相同.乙获胜.

yao4015

奇数应该是甲有必胜策略。
甲第一步只要不填第一，二行，然后再按上述策略就行了。

hbghlyj

如果要求填的数是1到$n^2$且不重复呢