Puzzleup 2017 (18) Three Circles - 拼圖

題目網址: http://www.puzzleup.com/2017/
http://www.puzzleup.com/2017/puzzle/?18
答題時限: 12月７日７ＰＭ－比賽結束（約12月27日）
加分時限: 12月７日７ＰＭ－12月12日６：５９ＰＭ

答對可得基本分１００分。答案可上傳５次，每改１次答案從基本分扣２０分。　
比賽期間內可隨時上傳答案，加分時限內答對第ｎ天加（６－ｎ）分　 　　 　　
另依題目的難易有額外加分（如有８０％的人這題答錯，答對者加８０分）　　

◆THREE CIRCLES

Two circles of the same size and different colors are placed randomly on a
computer screen. These circles can be in one of 6 different states:

1. The circles are not touching.
2. The circles are tangent, i.e. touching at a single point.
3. The circles are overlapping, the first one is on top.
4. The circles are overlapping, the second one is on top.
5. The circles are coincident, the first one is on top.
6. The circles are coincident, the second one is on top.

How many different states are there for 3 circles?

兩個相同大小但不同顏色的圓隨機畫在電腦螢幕畫面上。它們之間一共有六種狀況:

1. 兩圓不接觸
2. 兩圓相切，即它們在一點接觸
3. 兩圓有重疊，第一圓在上
4. 兩圓有重疊，第二圓在上
5. 兩圓重合，第一圓在上
6. 兩圓重合，第二圓在上

試問三圓共有幾種狀況？

Notes:
* If the order of overlapping or coincident circles are different in two
states, those states are considered different, even if they are visually the
same, e.g. because overlapping area is covered by another circle.
* The circles have a well defined order (top/bottom), e.g. if the first
circle is on top of the second and the second circle is on top of the third,
the third circle cannot be on top of the first.
* If two circles are coincident, the third one can be tangent to either both
of them or neither of them.

註:
* 只要重疊或重合的情形有所不同時即視為不同的狀況，
即使最後結果看起來是一樣的 (例如因為重疊處被第三圓蓋住了等等)。
(譯註: 舉個例子是上面的 3 4 兩種狀況，然後第三圓蓋住重疊區，這樣算兩種；
又例如三圓完全重合，這裡一共有六種，即使看起來只有三種)
* 三個圓之間是有上下順序關係的，
意即若第一圓在第二圓之上，第二圓在第三圓之上，則第三圓不可能在第一圓之上。
* 若兩圓重合，第三圓和兩圓相切的狀況可以是都相切或都不相切。
(譯註: 原文我看起來有點在強調不會只切其中一個)

--
有人喜歡邊玩遊戲邊上逼；
也有人喜歡邊聽歌邊打字。
但是，我有個請求，
選字的時候請專心好嗎？
-- 改編自「古 火田 任三郎」之開場白

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