抖音上一道有趣又很难的几何题

mathe

另外如果P3三格黄色映射到P1三个蓝色，但是P1两格黄色的像会分别来自P2和P3 (横向）,那么P2两个黄色的像也只能分别来自P2和P3(而且必须相邻），所以得出P1和P2的黄色格子只能构成正方形，同样P3和P2的蓝色格子也构成正方形，经试验，满足这些条件的图没有一个合法。
另外黄色格子构成5*1图案就两种，这时还会要求蓝色图案也出现5*1或1*5,穷举可以得到没有合法解。
而黄色构成4*1的图案会包含黄色L字形，于是要求对应的蓝色也有L字形，同样穷举没有合法解。
所以应该只有两个答案

倪举鹏

抖音上转直角的解确实是你解出来的两个，抖音上还有两种不是转直角的解，而且这里加了图案对称也算形状一样，可能解更多

mathe

如果允许图像部分自相交，的确还有，当然你这里规则没有说清楚。但是通常理解意义下，我们应该会不允许自相加交的。至于旋转角度不是直角倍数，显然不符合题意，即使中心坐标重叠也毫无意义。至于翻转，已经计算在内了。

mathe

为了方便描述各类格子，我们把P1中蓝色格子称为B1,黄色格子称为Y1. 同样如上图有B2,Y2,B3,Y3.
如果一个合法解对应的变换f将Y3变换为B1,Y1变换为B2, Y2变换为B3,那么我们称呼这种解为理想解。
而如果一个合法解对应的变换f将Y3变换为B1,但是Y1变换的结果不是B2或Y2变换结果不是B3,那么我们称呼这种解为准理想解。
而如果一个合法解对应的变换f将Y3变换的结果不是B1,那么，我们称呼这种解为异常解。
其中理想解是最容易解答的，也很容易计算。准理想解也相对容易分析，但是异常解分析会很复杂。但是从前面分析结果来看，很可能只有理想解。
楼主对于题目的描述有点不清楚，我们只能大概猜测一下题目中可能的扩展方向。
根据原题，我们会认为P1,P2,P3都只有水平和垂直两种方向放置方向，但是一种比较可能的扩展方案是我们可以不限制P1,P2,P3的放置方向，也就是它们都可以倾斜放置，那么应该就会容易有更多的解了。
其中P3我们还是可以认为5个格子的方向还是保持Y3在(0,-2),(0,-1),(0,0), 而B3在(0,1),(0,2)，设\(H=\begin{pmatrix}0, 1\\-1,0\end{pmatrix}\)代表逆时针旋转90度
可以假设变换f为 f(v)=Rv+a,其中R是一个二阶正交阵，a是一个二维向量。R可以是逆时针旋转t的角度的矩阵，也可以是选择加镜像的结果.
所以$R R'=I$, 其中\(R\begin{pmatrix}0\\ 1\end{pmatrix}\)代表B1的方向，于是\(HR\begin{pmatrix}0\\ 1\end{pmatrix}\)代表Y1的方向
所以\(RHR\begin{pmatrix}0\\ 1\end{pmatrix}\)代表B2也就是Y2的方向，\(R^2HR\begin{pmatrix}0\\ 1\end{pmatrix}\)代表B3也就是Y3的方向。
于是我们得出\(R^2HR\begin{pmatrix}0\\ 1\end{pmatrix}=\begin{pmatrix}0\\ 1\end{pmatrix}\)或\(R^2HR\begin{pmatrix}0\\ 1\end{pmatrix}=\begin{pmatrix}0\\ -1\end{pmatrix}\)
由于\(R^2HR\)是正交阵，根据几何意义，我们可以得出它只能是\(\begin{pmatrix}1,0\\ 0,1\end{pmatrix},\begin{pmatrix}-1,0\\ 0,1\end{pmatrix},\begin{pmatrix}1,0\\0, -1\end{pmatrix},\begin{pmatrix}-1,0\\ 0,-1\end{pmatrix}\)四者之一。

mathe

上面矩阵R可以分成两类，镜像对称或旋转变换。当R为镜像对称时，$R^2=I$,于是方程转化为$HR$为四者之一，于是只能\(R=\begin{pmatrix}0,1\\1,0\end{pmatrix}\)或\(R=\begin{pmatrix}0,-1\\-1,0\end{pmatrix}\),对应的变换还是将水平方向变化为垂直方向，并不会增加特别的解。
所以后面只需要讨论旋转变换即可

mathe

如果R代表旋转t度，那么等式表示旋转3t加90度后为180度倍数，于是除了正负90度还可以有正负30度和正负150度，后面就只余下在对应旋转角度下计算平移量的问题了，难度不大，但是有点繁琐

mathe

Found:
m=[0.86602540378443864676372317075293618347, -0.50000000000000000000000000000000000000; 0.50000000000000000000000000000000000000, 0.86602540378443864676372317075293618347]
vx=-0.68301270189221932338186158537646809176,vy=-0.45096189432334202985441524387059572479
A1=[-0.68301270189221932338186158537646809176, -0.45096189432334202985441524387059572479]:Blue
B1=[-0.18301270189221932338186158537646809176, -1.3169872981077806766181384146235319083]:Blue
C1=[0.68301270189221932338186158537646809171, -0.81698729810778067661813841462353190827]:Yellow
D1=[1.5490381056766579701455847561294042752, -0.31698729810778067661813841462353190827]:Yellow
E1=[0.31698729810778067661813841462353190824, -2.1830127018922193233818615853764680917]:Blue
A2=[0.31698729810778067661813841462353190822, -0.81698729810778067661813841462353190828]:Blue
B2=[0.81698729810778067661813841462353190824, 0.049038105676657970145584756129404275190]:Blue
C2=[1.8169872981077806766181384146235319083, 1.7810889132455352636730310976352766422]:Yellow
D2=[1.3169872981077806766181384146235319083, 0.91506350946109661690930792688234045867]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

B1和D3重叠，淘汰


Found:
m=[0.86602540378443864676372317075293618347, -0.50000000000000000000000000000000000000; 0.50000000000000000000000000000000000000, 0.86602540378443864676372317075293618347]
vx=0.049038105676657970145584756129404275213,vy=0.81698729810778067661813841462353190827
A1=[0.049038105676657970145584756129404275213, 0.81698729810778067661813841462353190827]:Blue
B1=[0.54903810567665797014558475612940427521, -0.049038105676657970145584756129404275202]:Blue
C1=[1.4150635094610966169093079268823404587, 0.45096189432334202985441524387059572480]:Yellow
D1=[2.2810889132455352636730310976352766422, 0.95096189432334202985441524387059572479]:Yellow
E1=[1.0490381056766579701455847561294042752, -0.91506350946109661690930792688234045867]:Blue
A2=[1.0490381056766579701455847561294042752, 1.9150635094610966169093079268823404587]:Blue
B2=[1.5490381056766579701455847561294042752, 2.7810889132455352636730310976352766422]:Blue
C2=[0.54903810567665797014558475612940427520, 1.0490381056766579701455847561294042752]:Yellow
D2=[0.049038105676657970145584756129404275186, 0.18301270189221932338186158537646809175]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

A1和C3重叠，淘汰



Found:
m=[0.86602540378443864676372317075293618347, -0.50000000000000000000000000000000000000; 0.50000000000000000000000000000000000000, 0.86602540378443864676372317075293618347]
vx=0.59807621135331594029116951225880855042,vy=1.7679491924311227064725536584941276331
A1=[0.59807621135331594029116951225880855042, 1.7679491924311227064725536584941276331]:Blue
B1=[1.0980762113533159402911695122588085504, 0.90192378864668405970883048774119144960]:Blue
C1=[0.23205080756887729352744634150587236694, 0.40192378864668405970883048774119144960]:Yellow
D1=[-0.63397459621556135323627682924706381653, -0.098076211353315940291169512258808550403]:Yellow
E1=[1.5980762113533159402911695122588085504, 0.035898384862245412945107316988255266127]:Blue
A2=[0.59807621135331594029116951225880855040, 2.2320508075688772935274463415058723670]:Blue
B2=[0.098076211353315940291169512258808550403, 1.3660254037844386467637231707529361835]:Blue
C2=[-0.40192378864668405970883048774119144960, 0.50000000000000000000000000000000000000]:Yellow
D2=[-0.90192378864668405970883048774119144961, -0.36602540378443864676372317075293618347]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

C1和C3重叠，淘汰


Found:
m=[0.86602540378443864676372317075293618347, -0.50000000000000000000000000000000000000; 0.50000000000000000000000000000000000000, 0.86602540378443864676372317075293618347]
vx=-0.13397459621556135323627682924706381653,vy=0.50000000000000000000000000000000000000
A1=[-0.13397459621556135323627682924706381653, 0.50000000000000000000000000000000000000]:Blue
B1=[0.36602540378443864676372317075293618347, -0.36602540378443864676372317075293618347]:Blue
C1=[-0.50000000000000000000000000000000000000, -0.86602540378443864676372317075293618347]:Yellow
D1=[-1.3660254037844386467637231707529361835, -1.3660254037844386467637231707529361835]:Yellow
E1=[0.86602540378443864676372317075293618347, -1.2320508075688772935274463415058723669]:Blue
A2=[-0.13397459621556135323627682924706381653, -0.49999999999999999999999999999999999999]:Blue
B2=[-0.63397459621556135323627682924706381653, -1.3660254037844386467637231707529361835]:Blue
C2=[0.86602540378443864676372317075293618347, 1.2320508075688772935274463415058723670]:Yellow
D2=[0.36602540378443864676372317075293618347, 0.36602540378443864676372317075293618348]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

D2和C3重叠，淘汰


Found:
m=[0.E-38, -1.0000000000000000000000000000000000000; 1.0000000000000000000000000000000000000, 0.E-38]
vx=5.0000000000000000000000000000000000000,vy=-1.0000000000000000000000000000000000000
A1=[5.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
B1=[6.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
C1=[6.0000000000000000000000000000000000000, 0.E-38]:Yellow
D1=[6.0000000000000000000000000000000000000, 1.0000000000000000000000000000000000000]:Yellow
E1=[7.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
A2=[5.0000000000000000000000000000000000000, 5.0000000000000000000000000000000000000]:Blue
B2=[4.0000000000000000000000000000000000000, 5.0000000000000000000000000000000000000]:Blue
C2=[3.0000000000000000000000000000000000000, 5.0000000000000000000000000000000000000]:Yellow
D2=[2.0000000000000000000000000000000000000, 5.0000000000000000000000000000000000000]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

即9#答案


Found:
m=[0.E-38, -1.0000000000000000000000000000000000000; 1.0000000000000000000000000000000000000, 0.E-38]
vx=1.0000000000000000000000000000000000000,vy=-1.0000000000000000000000000000000000000
A1=[1.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
B1=[2.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
C1=[2.0000000000000000000000000000000000000, 0.E-38]:Yellow
D1=[2.0000000000000000000000000000000000000, 1.0000000000000000000000000000000000000]:Yellow
E1=[3.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
A2=[1.0000000000000000000000000000000000000, 1.0000000000000000000000000000000000000]:Blue
B2=[0.E-38, 1.0000000000000000000000000000000000000]:Blue
C2=[3.0000000000000000000000000000000000000, 1.0000000000000000000000000000000000000]:Yellow
D2=[2.0000000000000000000000000000000000000, 1.0000000000000000000000000000000000000]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

B2和B3重叠淘汰


Found:
m=[0.E-38, -1.0000000000000000000000000000000000000; 1.0000000000000000000000000000000000000, 0.E-38]
vx=-2.0000000000000000000000000000000000000,vy=-1.0000000000000000000000000000000000000
A1=[-2.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
B1=[-1.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
C1=[-1.0000000000000000000000000000000000000, -2.0000000000000000000000000000000000000]:Yellow
D1=[-1.0000000000000000000000000000000000000, -3.0000000000000000000000000000000000000]:Yellow
E1=[0.E-37, -1.0000000000000000000000000000000000000]:Blue
A2=[0.E-38, -2.0000000000000000000000000000000000000]:Blue
B2=[1.0000000000000000000000000000000000000, -2.0000000000000000000000000000000000000]:Blue
C2=[3.0000000000000000000000000000000000000, -2.0000000000000000000000000000000000000]:Yellow
D2=[2.0000000000000000000000000000000000000, -2.0000000000000000000000000000000000000]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

E1和D3重叠，淘汰


Found:
m=[0.E-38, -1.0000000000000000000000000000000000000; 1.0000000000000000000000000000000000000, 0.E-38]
vx=2.0000000000000000000000000000000000000,vy=-1.0000000000000000000000000000000000000
A1=[2.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
B1=[3.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
C1=[3.0000000000000000000000000000000000000, -2.0000000000000000000000000000000000000]:Yellow
D1=[3.0000000000000000000000000000000000000, -3.0000000000000000000000000000000000000]:Yellow
E1=[4.0000000000000000000000000000000000000, -1.0000000000000000000000000000000000000]:Blue
A2=[4.0000000000000000000000000000000000000, 2.0000000000000000000000000000000000000]:Blue
B2=[5.0000000000000000000000000000000000000, 2.0000000000000000000000000000000000000]:Blue
C2=[3.0000000000000000000000000000000000000, 2.0000000000000000000000000000000000000]:Yellow
D2=[2.0000000000000000000000000000000000000, 2.0000000000000000000000000000000000000]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

等价1#答案


Found:
m=[-0.86602540378443864676372317075293618347, -0.50000000000000000000000000000000000000; 0.50000000000000000000000000000000000000, -0.86602540378443864676372317075293618347]
vx=0.18301270189221932338186158537646809171,vy=-3.0490381056766579701455847561294042752
A1=[0.18301270189221932338186158537646809171, -3.0490381056766579701455847561294042752]:Blue
B1=[0.68301270189221932338186158537646809171, -2.1830127018922193233818615853764680918]:Blue
C1=[-0.18301270189221932338186158537646809176, -1.6830127018922193233818615853764680918]:Yellow
D1=[-1.0490381056766579701455847561294042752, -1.1830127018922193233818615853764680918]:Yellow
E1=[1.1830127018922193233818615853764680917, -1.3169872981077806766181384146235319083]:Blue
A2=[1.1830127018922193233818615853764680918, -1.6830127018922193233818615853764680918]:Blue
B2=[1.6830127018922193233818615853764680917, -2.5490381056766579701455847561294042752]:Blue
C2=[2.6830127018922193233818615853764680917, -4.2810889132455352636730310976352766422]:Yellow
D2=[2.1830127018922193233818615853764680917, -3.4150635094610966169093079268823404587]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

C1和E3重叠，淘汰


Found:
m=[-0.86602540378443864676372317075293618347, -0.50000000000000000000000000000000000000; 0.50000000000000000000000000000000000000, -0.86602540378443864676372317075293618347]
vx=-2.5490381056766579701455847561294042752,vy=1.6830127018922193233818615853764680918
A1=[-2.5490381056766579701455847561294042752, 1.6830127018922193233818615853764680918]:Blue
B1=[-2.0490381056766579701455847561294042752, 2.5490381056766579701455847561294042752]:Blue
C1=[-2.9150635094610966169093079268823404587, 3.0490381056766579701455847561294042752]:Yellow
D1=[-3.7810889132455352636730310976352766422, 3.5490381056766579701455847561294042752]:Yellow
E1=[-1.5490381056766579701455847561294042752, 3.4150635094610966169093079268823404587]:Blue
A2=[-1.5490381056766579701455847561294042752, -2.4150635094610966169093079268823404587]:Blue
B2=[-1.0490381056766579701455847561294042752, -3.2810889132455352636730310976352766422]:Blue
C2=[-2.0490381056766579701455847561294042752, -1.5490381056766579701455847561294042752]:Yellow
D2=[-2.5490381056766579701455847561294042752, -0.68301270189221932338186158537646809175]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

这是一个合法解


Found:
m=[-0.86602540378443864676372317075293618347, -0.50000000000000000000000000000000000000; 0.50000000000000000000000000000000000000, -0.86602540378443864676372317075293618347]
vx=-4.5980762113533159402911695122588085504,vy=5.2320508075688772935274463415058723669
A1=[-4.5980762113533159402911695122588085504, 5.2320508075688772935274463415058723669]:Blue
B1=[-4.0980762113533159402911695122588085504, 6.0980762113533159402911695122588085504]:Blue
C1=[-3.2320508075688772935274463415058723670, 5.5980762113533159402911695122588085504]:Yellow
D1=[-2.3660254037844386467637231707529361835, 5.0980762113533159402911695122588085504]:Yellow
E1=[-3.5980762113533159402911695122588085504, 6.9641016151377545870548926830117447339]:Blue
A2=[-4.5980762113533159402911695122588085504, -1.2320508075688772935274463415058723670]:Blue
B2=[-5.0980762113533159402911695122588085504, -0.36602540378443864676372317075293618347]:Blue
C2=[-5.5980762113533159402911695122588085504, 0.50000000000000000000000000000000000000]:Yellow
D2=[-6.0980762113533159402911695122588085505, 1.3660254037844386467637231707529361835]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

这个也是合法解



Found:
m=[-0.86602540378443864676372317075293618347, -0.50000000000000000000000000000000000000; 0.50000000000000000000000000000000000000, -0.86602540378443864676372317075293618347]
vx=-1.8660254037844386467637231707529361835,vy=0.49999999999999999999999999999999999998
A1=[-1.8660254037844386467637231707529361835, 0.49999999999999999999999999999999999998]:Blue
B1=[-1.3660254037844386467637231707529361835, 1.3660254037844386467637231707529361835]:Blue
C1=[-0.50000000000000000000000000000000000002, 0.86602540378443864676372317075293618345]:Yellow
D1=[0.36602540378443864676372317075293618346, 0.36602540378443864676372317075293618344]:Yellow
E1=[-0.86602540378443864676372317075293618349, 2.2320508075688772935274463415058723669]:Blue
A2=[-1.8660254037844386467637231707529361835, -0.50000000000000000000000000000000000001]:Blue
B2=[-2.3660254037844386467637231707529361835, 0.36602540378443864676372317075293618347]:Blue
C2=[-0.86602540378443864676372317075293618348, -2.2320508075688772935274463415058723670]:Yellow
D2=[-1.3660254037844386467637231707529361835, -1.3660254037844386467637231707529361835]:Yellow
A3=[0, 2]:Blue
B3=[0, 1]:Blue
C3=[0, 0]:Yellow
D3=[0, -1]:Yellow
E3=[0, -2]:Yellow

C3和D1重叠淘汰

倪举鹏

4个解全部让你找出来了，是不是加了对称算一样的图的条件，并没有增加解的个数

倪举鹏

4个