這有公式嗎?! - 拼圖
By Sierra Rose
at 2016-02-10T19:18
at 2016-02-10T19:18
Table of Contents
要公式的話就從畢氏數組公式下手吧:
最簡畢氏數組可由 (m^2-n^2, 2mn, m^2+n^2) 產生, 其中 m n 互質且一奇一偶
因為 m^2+n^2 一定是三數裡最大的, 所以面積就是 mn(m^2-n^2)
考慮進 k 倍放大的話就是 k^2*mn(m+n)(m-n)
那麼就來列表啦:
m n mn(m+n)(m-n) m n mn(m+n)(m-n)
-------------------- --------------------
2 1 6 8 1 504
3 2 30 8 3 1320
4 1 60 8 5 1560
4 3 84 8 7 840
5 2 210 9 2 1386
5 4 180 9 4 2340
6 1 210 9 8 1224
6 5 330 10 1 990
7 2 630 10 3 2730
7 4 924 10 7 3570
7 6 546 10 9 1710
灰字就是連 k=1 面積都超過 1000 的
接下來也不是所有公倍數都能用, 要是 k^2 倍才行
所以再次列表:
m n k=1 2 3 4 5 ...
------------------------------
2 1 6 24 54 96 150 216 294 384 486 600 726 864 x
3 2 30 120 270 480 750 x
4 1 60 240 540 960 x
4 3 84 336 756 x
5 2 210 840 x
5 4 180 720 x
6 1 210 840 x
6 5 330 x
7 2 630 x
7 4 924 x
7 6 546 x
8 1 504 x
8 7 840 x
10 1 990 x
於是可以看到唯一出現三次的解即是表中黃字的數字 840
對應的三角形為: (m,n,k)=(5,2,2) => (42,40,58)
(m,n,k)=(6,1,2) => (70,24,74)
(m,n,k)=(8,7,1) => (15,112,113)
====
Bonus:
OEIS 裡有這麼一條數列: http://oeis.org/A177021
容易看到前面這些項都是 840 的倍數
不過並不都是上面這一組的整倍數
例如 10920 = 840*13, 對應的是
(m,n,k)=(10,3,2) => (182,120,218)
(m,n,k)=(13,8,1) => (105,208,233)
(m,n,k)=(14,1,2) => (390,56,694)
倍數也不一定像 13 一樣是平方和, 像 31920 = 840*38 這個解對應
(m,n,k)=(12,7,2) => (190,336,386)
(m,n,k)=(19,16,1) => (105,608,617)
(m,n,k)=(20,1,2) => (798,80,802)
然後也不是所有的面積都是 840 的倍數
此數列的說明文有提到一組 13123110 對應三個最簡畢氏數組:
(m,n,k)=(138,5,1) => (19019,1380,19069)
(m,n,k)=(78,55,1) => (3059,8580,9109)
(m,n,k)=(77,38,1) => (4485,5852,7373)
但 13123110 就不是 840 的倍數了
--
'Oh, Harry, don't you see?' Hermione breathed. 'If she could have done
one thing to make absolutely sure that every single person in this school
will read your interview, it was banning it!'
---'Harry Potter and the order of the phoenix', P513
--
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By Quintina
at 2016-02-15T15:12
at 2016-02-15T15:12
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