找回密码
 欢迎注册
楼主: 王守恩

[投票] 把1个正方形分成4个面积都是整数的三角形

[复制链接]
 楼主| 发表于 2023-1-13 15:10:27 | 显示全部楼层
谢谢 northwolves!

直角等腰(腰长是整数)三角形,在两条腰(整数位置)上作直线,两条直线相交于斜边,
连接腰上的2个点,这样把直角等腰三角形分成了4个三角形,要求4个三角形面积都是整数。

当腰长=n时,有a(n)种分法。还可以有吗?
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2023-1-13 21:41:12 | 显示全部楼层
本帖最后由 northwolves 于 2023-1-13 21:43 编辑

s1234.png 如图,
$s1=\frac{1}{2}xy$
$s2=\frac{1}{2}(n-x)(n-z)$
$s3=\frac{1}{2}(x(n-y)-z(x-y))$
$s4=\frac{1}{2}(n-y)z$

$1<=x,y,z<=n$

毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2023-1-13 21:41:50 | 显示全部楼层
本帖最后由 northwolves 于 2023-1-13 21:47 编辑
  1. def a(n):
  2.     t=[]
  3.     for x in range(1,n):
  4.         for y in range(x,n):
  5.             for z in range(1,n):
  6.                 s=[x*y,(n-x)*(n-z),x*(n-y)-z*(x-y),z*(n-y)]
  7.                 if all(s[k]%2==0 for k in range(4)):
  8.                     t.append([s[k]//2 for k in range(4)])
  9.     return(len(t),t)
  10. for n in range(1,11):
  11.     print(n,a(n))
复制代码


1 (0, [])
2 (0, [])
3 (0, [])
4 (5, [[1, 3, 2, 2], [2, 3, 2, 1], [2, 2, 2, 2], [2, 1, 2, 3], [3, 2, 2, 1]])
5 (0, [])
6 (27, [[1, 10, 3, 4], [1, 5, 4, 8], [2, 10, 4, 2], [2, 5, 7, 4], [2, 10, 4, 2], [2, 8, 4, 4], [2, 6, 4, 6], [2, 4, 4, 8], [2, 2, 4, 10], [3, 8, 4, 3], [3, 4, 5, 6], [4, 10, 3, 1], [4, 8, 4, 2], [4, 6, 5, 3], [4, 4, 6, 4], [4, 2, 7, 5], [5, 8, 4, 1], [5, 4, 7, 2], [6, 6, 4, 2], [6, 3, 5, 4], [8, 5, 4, 1], [8, 4, 4, 2], [8, 3, 4, 3], [8, 2, 4, 4], [8, 1, 4, 5], [10, 4, 3, 1], [10, 2, 4, 2]])
7 (0, [])
8 (78, [[1, 21, 4, 6], [1, 14, 5, 12], [1, 7, 6, 18], [2, 21, 5, 4], [2, 14, 8, 8], [2, 7, 11, 12], [3, 21, 6, 2], [3, 14, 11, 4], [3, 7, 16, 6], [2, 21, 6, 3], [2, 18, 6, 6], [2, 15, 6, 9], [2, 12, 6, 12], [2, 9, 6, 15], [2, 6, 6, 18], [2, 3, 6, 21], [3, 18, 6, 5], [3, 12, 7, 10], [3, 6, 8, 15], [4, 21, 5, 2], [4, 18, 6, 4], [4, 15, 7, 6], [4, 12, 8, 8], [4, 9, 9, 10], [4, 6, 10, 12], [4, 3, 11, 14], [5, 18, 6, 3], [5, 12, 9, 6], [5, 6, 12, 9], [6, 21, 4, 1], [6, 18, 6, 2], [6, 15, 8, 3], [6, 12, 10, 4], [6, 9, 12, 5], [6, 6, 14, 6], [6, 3, 16, 7], [7, 18, 6, 1], [7, 12, 11, 2], [7, 6, 16, 3], [6, 15, 7, 4], [6, 10, 8, 8], [6, 5, 9, 12], [9, 15, 6, 2], [9, 10, 9, 4], [9, 5, 12, 6], [8, 14, 8, 2], [8, 12, 8, 4], [8, 10, 8, 6], [8, 8, 8, 8], [8, 6, 8, 10], [8, 4, 8, 12], [8, 2, 8, 14], [10, 12, 7, 3], [10, 8, 8, 6], [10, 4, 9, 9], [12, 14, 5, 1], [12, 12, 6, 2], [12, 10, 7, 3], [12, 8, 8, 4], [12, 6, 9, 5], [12, 4, 10, 6], [12, 2, 11, 7], [14, 12, 5, 1], [14, 8, 8, 2], [14, 4, 11, 3], [15, 9, 6, 2], [15, 6, 7, 4], [15, 3, 8, 6], [18, 7, 6, 1], [18, 6, 6, 2], [18, 5, 6, 3], [18, 4, 6, 4], [18, 3, 6, 5], [18, 2, 6, 6], [18, 1, 6, 7], [21, 6, 4, 1], [21, 4, 5, 2], [21, 2, 6, 3]])
9 (0, [])
10 (170, [[1, 36, 5, 8], [1, 27, 6, 16], [1, 18, 7, 24], [1, 9, 8, 32], [2, 36, 6, 6], [2, 27, 9, 12], [2, 18, 12, 18], [2, 9, 15, 24], [3, 36, 7, 4], [3, 27, 12, 8], [3, 18, 17, 12], [3, 9, 22, 16], [4, 36, 8, 2], [4, 27, 15, 4], [4, 18, 22, 6], [4, 9, 29, 8], [2, 36, 8, 4], [2, 32, 8, 8], [2, 28, 8, 12], [2, 24, 8, 16], [2, 20, 8, 20], [2, 16, 8, 24], [2, 12, 8, 28], [2, 8, 8, 32], [2, 4, 8, 36], [3, 32, 8, 7], [3, 24, 9, 14], [3, 16, 10, 21], [3, 8, 11, 28], [4, 36, 7, 3], [4, 32, 8, 6], [4, 28, 9, 9], [4, 24, 10, 12], [4, 20, 11, 15], [4, 16, 12, 18], [4, 12, 13, 21], [4, 8, 14, 24], [4, 4, 15, 27], [5, 32, 8, 5], [5, 24, 11, 10], [5, 16, 14, 15], [5, 8, 17, 20], [6, 36, 6, 2], [6, 32, 8, 4], [6, 28, 10, 6], [6, 24, 12, 8], [6, 20, 14, 10], [6, 16, 16, 12], [6, 12, 18, 14], [6, 8, 20, 16], [6, 4, 22, 18], [7, 32, 8, 3], [7, 24, 13, 6], [7, 16, 18, 9], [7, 8, 23, 12], [8, 36, 5, 1], [8, 32, 8, 2], [8, 28, 11, 3], [8, 24, 14, 4], [8, 20, 17, 5], [8, 16, 20, 6], [8, 12, 23, 7], [8, 8, 26, 8], [8, 4, 29, 9], [9, 32, 8, 1], [9, 24, 15, 2], [9, 16, 22, 3], [9, 8, 29, 4], [6, 28, 10, 6], [6, 21, 11, 12], [6, 14, 12, 18], [6, 7, 13, 24], [9, 28, 9, 4], [9, 21, 12, 8], [9, 14, 15, 12], [9, 7, 18, 16], [12, 28, 8, 2], [12, 21, 13, 4], [12, 14, 18, 6], [12, 7, 23, 8], [8, 27, 12, 3], [8, 24, 12, 6], [8, 21, 12, 9], [8, 18, 12, 12], [8, 15, 12, 15], [8, 12, 12, 18], [8, 9, 12, 21], [8, 6, 12, 24], [8, 3, 12, 27], [10, 24, 11, 5], [10, 18, 12, 10], [10, 12, 13, 15], [10, 6, 14, 20], [12, 27, 9, 2], [12, 24, 10, 4], [12, 21, 11, 6], [12, 18, 12, 8], [12, 15, 13, 10], [12, 12, 14, 12], [12, 9, 15, 14], [12, 6, 16, 16], [12, 3, 17, 18], [14, 24, 9, 3], [14, 18, 12, 6], [14, 12, 15, 9], [14, 6, 18, 12], [16, 27, 6, 1], [16, 24, 8, 2], [16, 21, 10, 3], [16, 18, 12, 4], [16, 15, 14, 5], [16, 12, 16, 6], [16, 9, 18, 7], [16, 6, 20, 8], [16, 3, 22, 9], [18, 24, 7, 1], [18, 18, 12, 2], [18, 12, 17, 3], [18, 6, 22, 4], [15, 20, 11, 4], [15, 15, 12, 8], [15, 10, 13, 12], [15, 5, 14, 16], [20, 20, 8, 2], [20, 15, 11, 4], [20, 10, 14, 6], [20, 5, 17, 8], [18, 18, 12, 2], [18, 16, 12, 4], [18, 14, 12, 6], [18, 12, 12, 8], [18, 10, 12, 10], [18, 8, 12, 12], [18, 6, 12, 14], [18, 4, 12, 16], [18, 2, 12, 18], [21, 16, 10, 3], [21, 12, 11, 6], [21, 8, 12, 9], [21, 4, 13, 12], [24, 18, 7, 1], [24, 16, 8, 2], [24, 14, 9, 3], [24, 12, 10, 4], [24, 10, 11, 5], [24, 8, 12, 6], [24, 6, 13, 7], [24, 4, 14, 8], [24, 2, 15, 9], [27, 16, 6, 1], [27, 12, 9, 2], [27, 8, 12, 3], [27, 4, 15, 4], [28, 12, 8, 2], [28, 9, 9, 4], [28, 6, 10, 6], [28, 3, 11, 8], [32, 9, 8, 1], [32, 8, 8, 2], [32, 7, 8, 3], [32, 6, 8, 4], [32, 5, 8, 5], [32, 4, 8, 6], [32, 3, 8, 7], [32, 2, 8, 8], [32, 1, 8, 9], [36, 8, 5, 1], [36, 6, 6, 2], [36, 4, 7, 3], [36, 2, 8, 4]])

评分

参与人数 1威望 +26 金币 +26 贡献 +26 经验 +26 鲜花 +26 收起 理由
王守恩 + 26 + 26 + 26 + 26 + 26 千言万语并一句:新年快乐!

查看全部评分

毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2023-1-13 21:59:31 | 显示全部楼层
$a(n)=\frac{(1+(-1)^n)*n*(n-2)*(2*n-3)}{16} $

显然单数项均为0

前200项:

[0, 0, 0, 5, 0, 27, 0, 78, 0, 170, 0, 315, 0, 525, 0, 812, 0, 1188, 0, 1665, 0, 2255, 0, 2970, 0, 3822, 0, 4823, 0, 5985, 0, 7320, 0, 8840, 0, 10557, 0, 12483, 0, 14630, 0, 17010, 0, 19635, 0, 22517, 0, 25668, 0, 29100, 0, 32825, 0, 36855, 0, 41202, 0, 45878, 0, 50895, 0, 56265, 0, 62000, 0, 68112, 0, 74613, 0, 81515, 0, 88830, 0, 96570, 0, 104747, 0, 113373, 0, 122460, 0, 132020, 0, 142065, 0, 152607, 0, 163658, 0, 175230, 0, 187335, 0, 199985, 0, 213192, 0, 226968, 0, 241325, 0, 256275, 0, 271830, 0, 288002, 0, 304803, 0, 322245, 0, 340340, 0, 359100, 0, 378537, 0, 398663, 0, 419490, 0, 441030, 0, 463295, 0, 486297, 0, 510048, 0, 534560, 0, 559845, 0, 585915, 0, 612782, 0, 640458, 0, 668955, 0, 698285, 0, 728460, 0, 759492, 0, 791393, 0, 824175, 0, 857850, 0, 892430, 0, 927927, 0, 964353, 0, 1001720, 0, 1040040, 0, 1079325, 0, 1119587, 0, 1160838, 0, 1203090, 0, 1246355, 0, 1290645, 0, 1335972, 0, 1382348, 0, 1429785, 0, 1478295, 0, 1527890, 0, 1578582, 0, 1630383, 0, 1683305, 0, 1737360, 0, 1792560, 0, 1848917, 0, 1906443, 0, 1965150]
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2023-1-14 13:35:42 | 显示全部楼层
谢谢 northwolves!我已经跟不上你的方法了,但还是想把题目串成一串。

直角(直角边是整数)三角形,在两条直角边(整数位置)上作直线,两条直线相交于斜边,
连接直角边上的2个点,这样把直角三角形分成了4个三角形,要求4个三角形面积都是整数。

当较长直角边(相等也可以)=n时,有a(n)种分法。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2023-1-14 22:47:30 | 显示全部楼层
王守恩 发表于 2023-1-14 13:35
谢谢 northwolves!我已经跟不上你的方法了,但还是想把题目串成一串。

直角(直角边是整数)三角形,在两 ...
  1. def a(n):
  2.     t=[]
  3.     for m in range(1,n+1):
  4.         r=int((m*m+n*n)**0.5)
  5.         for x in range(1,n):
  6.             for y in range(1,m):
  7.                 for k in range(1,r):
  8.                     s=[x*y,(n-x)*m*(1-k/r),k/r*(n*y-m*x)+x*(m-y),(m-y)*n*k/r]
  9.                     if all(s[k]%2==0 for k in range(4)):
  10.                         t.append([int(s[k]/2) for k in range(4)])
  11.     return(len(t),t)
  12. for n in range(1,11):
  13.     print(n,a(n))
复制代码


1 (0, [])
2 (0, [])
3 (0, [])
4 (1, [[1, 1, 1, 1]])
5 (1, [[1, 2, 2, 5]])
6 (17, [[1, 5, 2, 1], [1, 4, 2, 2], [1, 3, 2, 3], [1, 2, 2, 4], [2, 4, 2, 1], [2, 2, 3, 2], [2, 2, 3, 2], [2, 1, 2, 4], [4, 2, 2, 1], [4, 1, 2, 2], [5, 1, 2, 1], [2, 9, 4, 3], [2, 6, 4, 6], [2, 3, 4, 9], [4, 6, 5, 3], [4, 3, 5, 6], [8, 3, 4, 3]])
7 (1, [[9, 1, 4, 7]])
8 (65, [[1, 3, 2, 2], [2, 3, 2, 1], [2, 2, 2, 2], [2, 1, 2, 3], [3, 1, 2, 2], [2, 3, 3, 4], [4, 3, 3, 2], [1, 7, 4, 4], [1, 9, 3, 3], [1, 6, 3, 6], [1, 3, 3, 9], [2, 9, 3, 2], [2, 6, 4, 4], [2, 3, 5, 6], [3, 9, 3, 1], [3, 6, 5, 2], [3, 3, 7, 3], [3, 5, 4, 4], [2, 6, 5, 3], [2, 4, 4, 6], [2, 2, 3, 9], [4, 7, 4, 1], [4, 6, 4, 2], [4, 5, 4, 3], [4, 4, 4, 4], [4, 3, 4, 5], [4, 2, 4, 6], [4, 1, 4, 7], [6, 6, 3, 1], [6, 4, 4, 2], [6, 2, 5, 3], [5, 3, 4, 4], [3, 3, 7, 3], [3, 2, 5, 6], [3, 1, 3, 9], [6, 3, 5, 2], [6, 2, 4, 4], [6, 1, 3, 6], [9, 3, 3, 1], [9, 2, 3, 2], [9, 1, 3, 3], [7, 1, 4, 4], [1, 9, 4, 10], [2, 9, 5, 8], [3, 9, 6, 6], [4, 9, 7, 4], [5, 9, 8, 2], [2, 6, 6, 10], [4, 6, 6, 8], [6, 6, 6, 6], [8, 6, 6, 4], [10, 6, 6, 2], [3, 3, 8, 10], [6, 3, 7, 8], [9, 3, 6, 6], [12, 3, 5, 4], [15, 3, 4, 2], [3, 14, 7, 4], [3, 7, 6, 12], [2, 7, 7, 12], [4, 7, 7, 10], [6, 7, 7, 8], [8, 7, 7, 6], [10, 7, 7, 4], [12, 7, 7, 2]])
9 (38, [[3, 1, 2, 3], [3, 8, 4, 3], [3, 4, 5, 6], [3, 2, 4, 9], [6, 4, 5, 3], [6, 2, 4, 6], [9, 2, 4, 3], [1, 12, 5, 9], [3, 9, 6, 9], [2, 9, 7, 9], [5, 6, 7, 9], [7, 3, 8, 9], [2, 16, 9, 9], [4, 14, 9, 9], [3, 16, 8, 9], [3, 8, 7, 18], [6, 20, 7, 3], [6, 16, 8, 6], [6, 12, 9, 9], [6, 8, 10, 12], [9, 16, 8, 3], [9, 8, 13, 6], [8, 10, 9, 9], [10, 8, 9, 9], [3, 4, 8, 21], [6, 8, 13, 9], [6, 4, 8, 18], [9, 4, 8, 15], [12, 10, 11, 3], [12, 8, 10, 6], [12, 6, 9, 9], [12, 4, 8, 12], [15, 4, 8, 9], [18, 8, 7, 3], [18, 4, 8, 6], [21, 4, 8, 3], [14, 4, 9, 9], [16, 2, 9, 9]])
10 (96, [[1, 6, 3, 5], [5, 6, 3, 1], [5, 3, 4, 3], [3, 3, 4, 5], [1, 9, 5, 5], [2, 8, 5, 5], [3, 7, 5, 5], [4, 6, 5, 5], [5, 9, 5, 1], [5, 8, 5, 2], [5, 7, 5, 3], [5, 6, 5, 4], [5, 5, 5, 5], [5, 4, 5, 6], [6, 4, 5, 5], [7, 3, 5, 5], [8, 2, 5, 5], [9, 1, 5, 5], [1, 14, 5, 15], [3, 21, 6, 5], [3, 14, 8, 10], [3, 7, 10, 15], [5, 14, 11, 5], [3, 7, 10, 15], [9, 7, 9, 10], [15, 7, 8, 5], [1, 30, 4, 5], [1, 18, 6, 15], [2, 27, 6, 5], [2, 18, 10, 10], [2, 9, 14, 15], [3, 18, 14, 5], [2, 16, 7, 15], [4, 24, 7, 5], [4, 16, 10, 10], [4, 8, 13, 15], [6, 16, 13, 5], [3, 14, 8, 15], [6, 21, 8, 5], [6, 14, 10, 10], [6, 7, 12, 15], [9, 14, 12, 5], [4, 20, 11, 5], [4, 16, 10, 10], [4, 12, 9, 15], [4, 8, 8, 20], [8, 18, 9, 5], [8, 12, 10, 10], [8, 6, 11, 15], [10, 16, 9, 5], [10, 8, 12, 10], [12, 12, 11, 5], [5, 10, 10, 15], [10, 15, 10, 5], [10, 10, 10, 10], [10, 5, 10, 15], [15, 10, 10, 5], [6, 8, 11, 15], [12, 12, 11, 5], [12, 8, 10, 10], [12, 4, 9, 15], [18, 8, 9, 5], [7, 10, 18, 5], [7, 8, 15, 10], [7, 6, 12, 15], [7, 4, 9, 20], [14, 9, 12, 5], [14, 6, 10, 10], [14, 3, 8, 15], [21, 6, 8, 5], [8, 4, 13, 15], [16, 6, 13, 5], [16, 4, 10, 10], [16, 2, 7, 15], [24, 4, 7, 5], [9, 2, 14, 15], [18, 3, 14, 5], [18, 2, 10, 10], [18, 1, 6, 15], [27, 2, 6, 5], [2, 20, 8, 20], [4, 20, 11, 15], [6, 20, 14, 10], [8, 20, 17, 5], [4, 15, 11, 20], [8, 15, 12, 15], [12, 15, 13, 10], [16, 15, 14, 5], [6, 10, 14, 20], [12, 10, 13, 15], [18, 10, 12, 10], [24, 10, 11, 5], [8, 5, 17, 20], [16, 5, 14, 15], [24, 5, 11, 10], [32, 5, 8, 5]])

评分

参与人数 1威望 +36 金币 +36 贡献 +36 经验 +36 鲜花 +36 收起 理由
王守恩 + 36 + 36 + 36 + 36 + 36 新年快乐!给您发红包是必须的。

查看全部评分

毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2023-1-14 22:51:58 | 显示全部楼层
a(1)-a(60):

[0, 0, 0, 1, 1, 17, 1, 65, 38, 96, 17, 353, 32, 241, 128, 838, 118, 1113, 209, 2114, 672, 624, 414, 3421, 752, 1958, 1906, 7357, 886, 4326, 785, 10314, 2212, 4561, 3706, 16764, 2092, 5665, 3280, 23540, 3152, 14479, 2960, 24951, 13664, 13374, 4843, 31014, 10459, 30418, 8654, 42192, 6187, 38914, 12088, 66777, 9711, 25624, 7419, 89841]

暂未找到通项公式
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2023-1-14 23:04:08 | 显示全部楼层
1111.png 如图,$AC=m,CB=n,AB=r$

$S_1=\frac{1}{2}xy$
$S_2=\frac{1}{2}(n-x)(r-z)*m/r$
$S_4=\frac{1}{2}(m-y)*z*n/r$
$S_3=\frac{1}{2}(mn-xy-(n-x)(r-z)*m/r-(m-y)*z*n/r)$
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2023-1-14 23:24:45 | 显示全部楼层
本帖最后由 northwolves 于 2023-1-14 23:29 编辑

$r/z不一定是整数,26,27楼的数据有漏解的可能。$
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2023-1-17 13:05:35 | 显示全部楼层
终极版。

任意(较长边是整数)三角形,在三条边(较长边在整数位置)上作直线,
两两相交,这样把三角形分成了4个三角形,要求4个三角形面积都是整数。

当较长边(相等也可以)=n时,有a(n)种分法。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
您需要登录后才可以回帖 登录 | 欢迎注册

本版积分规则

小黑屋|手机版|数学研发网 ( 苏ICP备07505100号 )

GMT+8, 2024-3-29 07:48 , Processed in 0.095573 second(s), 21 queries .

Powered by Discuz! X3.5

© 2001-2024 Discuz! Team.

快速回复 返回顶部 返回列表