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# [投票] 把1个正方形分成4个面积都是整数的三角形

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04有02组: 4,4,2, 2,4,3,
05有02组:
06有03组: 4,6,1,
07有03组:
08有04组: 6,6,8,
09有05组: 6,9,6,
10有06组: 3,6,10,
11有06组:
12有10组: 12,12,2, 9,12,3, 6,12,4, 3,12,5,
13有10组:
14有10组:
15有13组: 12,15,1, 06,15,02, 8,12,15,
16有17组: 16,16,8, 12,16,10, 8,16,12, 4,16,14,

2, 2, 3, 3, 4, 5, 6, 6, 10, 10, 10, 13, 17, 17, 18, 18, 21, 23, 23, 23, 30, 32, 32, 33, 37, 37, 44, 44, 46, 46, 46, 50,
58, 58, 58, 58, 67, 67, 73, 73, 75, 81, 81, 81, 90, 93, 96, 96, 97, 97, 102, 105, 115, 115, 115, 115, 127, ......

 a(9)=8,  [[2, 4, 3], [2, 9, 8], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [8, 9, 5]]

 a(9)=8, [[2, 4, 3], [2, 9, 8], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [8, 9, 5]]

 a(4)=2:  [[2, 4, 3], [4, 4, 2]] a(5)=2:  [[2, 4, 3], [4, 4, 2]] a(6)=3:  [[2, 4, 3], [4, 4, 2], [4, 6, 1]] a(7)=3:  [[2, 4, 3], [4, 4, 2], [4, 6, 1]] a(8)=4:  [[2, 4, 3], [4, 4, 2], [4, 6, 1], [6, 6, 8]] a(9)=8:  [[2, 4, 3], [2, 9, 8], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [8, 9, 5]] a(10)=10:  [[2, 4, 3], [2, 9, 8], [3, 6, 10], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [8, 9, 5], [9, 10, 4]] a(11)=10:  [[2, 4, 3], [2, 9, 8], [3, 6, 10], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [8, 9, 5], [9, 10, 4]] a(12)=14:  [[2, 4, 3], [2, 9, 8], [3, 6, 10], [3, 12, 5], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [6, 12, 4], [8, 9, 5], [9, 10, 4], [9, 12, 3], [12, 12, 2]] a(13)=14:  [[2, 4, 3], [2, 9, 8], [3, 6, 10], [3, 12, 5], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [6, 12, 4], [8, 9, 5], [9, 10, 4], [9, 12, 3], [12, 12, 2]] a(14)=15:  [[2, 4, 3], [2, 9, 8], [3, 6, 10], [3, 12, 5], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [6, 12, 4], [8, 9, 5], [9, 10, 4], [9, 12, 3], [9, 14, 1], [12, 12, 2]] a(15)=18:  [[2, 4, 3], [2, 9, 8], [3, 6, 10], [3, 12, 5], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [6, 12, 4], [6, 15, 2], [8, 9, 5], [8, 12, 15], [9, 10, 4], [9, 12, 3], [9, 14, 1], [12, 12, 2], [12, 15, 1]] a(16)=27:  [[2, 4, 3], [2, 9, 8], [2, 16, 15], [3, 6, 10], [3, 12, 5], [4, 4, 2], [4, 6, 1], [4, 9, 7], [4, 16, 14], [6, 6, 8], [6, 9, 6], [6, 12, 4], [6, 15, 2], [6, 16, 13], [8, 9, 5], [8, 12, 15], [8, 16, 12], [9, 10, 4], [9, 12, 3], [9, 14, 1], [9, 16, 1], [10, 16, 11], [12, 12, 2], [12, 15, 1], [12, 16, 10], [14, 16, 9], [16, 16, 8]]

楼主| 发表于 2022-12-28 08:48:35 | 显示全部楼层
 本帖最后由 王守恩 于 2022-12-28 15:30 编辑 northwolves 发表于 2022-12-27 22:33 a(9)=8, [[2, 4, 3], [2, 9, 8], [4, 4, 2], [4, 6, 1], [4, 9, 7], [6, 6, 8], [6, 9, 6], [8, 9, 5]] 题目有缺陷。不过我是希望：这2个点是在整数位置上。且正方形边长是整数。

 假设正方形边长为a, 两个包含完整正方形边长的三角形另外一条直角边分别为x,y,于是要求 ax,ay都是偶数而且$(a-x)(a-y)=a^2-ax-ay+xy$也是偶数. 由于$a^2$也是整数，所以得到xy也是整数,而且$a^2,xy$同奇偶。 另外$a^2\times xy=ax\times ay$是4的倍数，所以$a^2, xy$都是偶数。 比如我们选择$a^2=18,xy=2, ax=ay=6$,也就是$a=3\sqrt{2}, x=y=\sqrt{2}$,可以得到三个三角形面积分别为3,3,4. 一般的，如果对于任意一个乘法分解mn=st,其中假设t是四个数中最大的。 我们总可以选择$ax=2m,ay=2n,a^2=2t, xy=2s$，得到三个三角形面积为$m,n,t+s-m-n$ 对应的解为$a=\sqrt{2t}, x=\frac{2m}{\sqrt{2t}},y=\frac{2n}{\sqrt{2t}}$

### 点评

 mathe 发表于 2022-12-28 08:51 假设正方形边长为a, 两个包含完整正方形边长的三角形另外一条直角边分别为x,y,于是要求 ax,ay都是偶数而且 ... 我的计算里也漏了好多解

 找一串看上去有些意义但是oeis没有收集的序列太容易了，比如 5 8 5 9 8 7 4 4 8 2 0 4 8 8 3 8 ,竟然被收集了，那就 1,0,5,3,0,6,4,8,7,5,2,5,2,0这下就没有了

楼主| 发表于 2022-12-29 18:34:14 | 显示全部楼层
 把1个正方形(边长是整数)分成(分点在整数位置上)4个面积都是整数的三角形，周边3个面积最大者不超过 04有02组: 4,4,2, 2,4,3, 05有02组: 06有03组: 4,6,1, 07有03组: 08有04组: 6,6,8, 09有05组: 6,9,6, 10有06组: 3,6,10, 11有06组: 12有10组: 12,12,2, 9,12,3, 6,12,4, 3,12,5, 13有10组: 14有10组: 15有13组: 12,15,1, 06,15,02, 8,12,15, 16有17组: 16,16,8, 12,16,10, 8,16,12, 4,16,14, 第3个数表示2边都不在完整边的那个三角形。 得到这样一串数: 2, 2, 3, 3, 4, 5, 6, 6, 10, 10, 10, 13, 17, 17, 18, 18, 21, 23, 23, 23, 30, 32, 32, 33, 37, 37, 44, 44, 46, 46, 46, 50, 58, 58, 58, 58, 67, 67, 73, 73, 75, 81, 81, 81, 90, 93, 96, 96, 97, 97, 102, 105, 115, 115, 115, 115, 127, ...... (1),Table[Table[Table[Table[Solve[{a*x == 2 A, b*x == 2 B, (x - a) (x - b) == 2 C, 0 < a <= b < x}, {a, b, x}, Integers], {C, 1, n}], {B, A, n}], {A, 1, n}], {n, 4, 4}] (2),Table[Table[Table[Table[(1/4)Dimensions[Solve[{a*x == 2 A, b*x == 2 B, (x - a) (x - b) == 2 C, 0 < a <= b < x}, {a, b, x}, Integers]], {C, 1, n}], {B, A, n}], {A, 1, n}] // Flatten // Total, {n, 4, 80}] 用(1)可以把[4,4,2], [2,4,3],[4,6,1],[6,6,8],[6,9,6],[3,6,10],......一个一个找出来， 用(2)可以把 2, 2, 3, 3, 4, 5, 6, 6, 10, 10, 10, 13, 17, 17, 18, 18, 21, 23, 23, 23, 30, 32, 32, 33, 37, 37, 44, 44, 46, 46, 46, 50, 58, 58, 58, 58, 67, 67, 73, 73, 75, 81, 81, 81, 90, 93, 96, 96, 97, 97, 102, 105, 115, 115, 115, 115, 127,......一个一个找出来， (1),(2)速度太慢了，真诚求助: (1),(2)可以提速吗？

楼主| 发表于 2023-1-6 11:04:23 | 显示全部楼层
 把1个矩形(边长是整数)分成(分点在整数位置上)4个面积都是整数(>1)的三角形，周边3个面积(不同)最大者不超过 4有01组: 2,4,3, 5有03组: 4,5,3, 2,5,4, 6有07组: 3,6,2, 5,6,2, 2,6,5, 3,4,6, 7有10组: 6,7,4, 4,7,5, 2,7,6, 8有16组: 6,8,2, 7,8,3, 6,8,5, 4,8,6, 2,8,7, 3,5,8, 9有23组: 4,9,2, 5,9,4, 8,9,5, 4,9,7, 2,9,8, 7,9,8, 5,8,9, 得到这样一串数: 1, 3, 7, 10, 16, 23, 36, 41, 60, 66, 85, 108, 132, 140, 166, 175, ...... 这可是一串在《整数序列在线百科全书(OEIS)》找不到的。

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