最少蓋片面積

2008-12-21 3:42 pm
一般遊樂場裡有一種攤位遊戲, 玩法為在一黑盒的底部有一圓形的白光面, 然後參加者會得到數塊黑圓片如下:

圖片參考:http://i117.photobucket.com/albums/o61/billy_hywung/Dec08/Crazygeom2.jpg

其中 r < R
參加者需要將數塊圓片放入黑盒中, 務求將整個白底完全遮蓋方為勝出.
問題為:
若果有 N 塊黑圓片時, 則每塊黑圓片的半徑 r 至少要多少才能完全遮蓋整個白底?
將答案以 N 和 R 表示.
當然, 當 N = 1 時, 好明顯 r 最少必為 R. 而當 N = 2 時, r 並非最少為 R/√2 如此簡單. 不信的話閣下可以嘗試以兩塊半徑為 R/√2 的黑圓片去遮蓋一半徑為 R 的圓白底, 就可以發現無論如何擺法皆不能完全遮蓋.

回答 (6)

2009-01-03 6:57 am
✔ 最佳答案
究竟黑盒壁是緊點白光面的邊界,還是與白光面的邊界留有空間?

2009-01-02 22:57:56 補充:

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參考資料:
my wisdom

2010-07-20 02:40:42 補充:
猛省,其實這題與http://www2.stetson.edu/~efriedma/circovcir很有關係,點解我當初搵唔到這個網站?

不過當然仍然都是沒有公式(事實上根據沒有可能找到公式),而且直到目前為止找到準確值的最大N值仍只是N = 12。
2009-01-04 5:32 am
恐怕N和R的關係表示式會因N的不同而不同,很難甚至根本不能得出一個統一的公式,這類問題一般都很棘手,甚至專家亦不能完全解釋。
2009-01-02 4:30 am
黑圓片可否重疊?
2008-12-27 4:32 am
是白光面的邊界留有空間, 而且可以假設足夠地大.
即蓋片的邊可以越過白光面的邊界, 就算成塊出左都得.

2009-01-02 00:27:48 補充:
當然可以, 否則就不可能完全遮蓋
2008-12-23 11:25 pm
我認為當 N = 2 時, r 最小都是R
點證便唔知,因為我點放都放唔到
2008-12-23 10:50 pm
無可能完全遮蓋整個白底


收錄日期: 2021-04-21 22:01:26
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20081221000051KK00284

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