'幫我翻譯一段英文(不要翻譯軟體)

Let (A; B) be any pair of friends. Remove this pair from consideration and for each pair

(C; D), where C is a friend of A and D is a friend of B, change the relationship between

C and D to the opposite; do the same if C is a friend of B and D is a friend of A. Note

that if C and D are friends of both A and B, their relationship will be reversed twice,

leaving it unchanged.

Consider now an arbitrary participant X di®erent from A and B and choose one of the two

dining rooms. [Note that in the case under consideration, the total number of participants

is at least 3, so such a triplet (A; B; X) can be chosen.] Let A have m friends in this

room and let B have n friends in this room; both m and n are even. When the pair

(A; B) is removed, X's relationship will be reversed with either n, or m, or m + n ¡ 2k

(for k the number of mutual friends of A and B in the chosen room), or 0 people within

the chosen room (depending on whether he/she is a friend of only A, only B, both, or

neither). Since m and n are both even, the parity of the number of X's friends in that

room will be therefore unchanged in any case.

Again, a legitimate seating without A and B will arise from exactly one arrangement that

includes the pair (A; B): just add each of A and B to the room with an odd number

of the other's friends, and then reverse all of the relationships between a friend of A and

a friend of B. In this way we create a one-to-one correspondence between all possible

seatings before and after the (A; B) removal.

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