Advanced Maths Questions

都係試下做, 做到幾多得幾多

1. Seven kids are numbered as 1, 2, 3, ... 7 are sitting on seven chairs which are numbered as 1, 2, 3, ..., 7 respectively. (i.e. Kid 1 sits on Chair 1, Kid 2 sits on Chair 2, ...) Now, they have to change their seats according to the following instructions:
(a) The kid should move. (i.e. Kid 1 cannot stay on his chair.)
(b) They are not allowed to directly interchange seats with each other.
(i. e. If Kid 1 sits on Chair 2, Kid 2 must not sit on Chair 1.
However, it is accepted that Kid 1 sits on Chair 2, Kid 2 sits on Chair 5 and Kid 5 sits on Chair 1.)

How many possible combinations are there after the change?

七個編碼為1, 2, 3, ..., 7的小孩依次地坐在編碼為1,2,3, ..., 7的座椅上。現在他們須依以下規例更改自己的座位︰
(a) 每位小孩必須更變座位。 (例如小孩1不能停留在座椅1上)
(b) 兩位小孩不能直接交換互相的座位。
(如小孩1坐在座椅2上,小孩2就不能坐在座椅1上 。可是, 小孩1坐在椅2,小孩2坐在椅5,小孩5坐在椅1,就可以被接納。)

求七位小孩更變後共有多少種坐法。

2. An indefinite arithmetic sequence has only positive integral terms. If none of the terms is a Fibonacci number (1, 2, 3, 5, 8, 13, 21, 34, 55, ...) , find the minimum of the common difference.

一個有無限項只包含正整數的算術(等差)數列, 沒有其中一項為Fibonacci number(即1,2, 3, 5, 8, 13, 21, 34, 55, ...), 求公差的最小值。