|---|---|---|---|---|
| a | | | b | |
| | | | | |
| 2 | 1 | | 1 | |
| 1 | | | 2 | 1 |
随机用结构a或者b填充平面，让两个结构只能以2-1-2的方式连接，让平面的尺寸是4*6，计算平均密度
不考虑顺序有4种可能，
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 1 | 2 | 1 | 2 | 1 | | 2 | 1 | | | 2 | 1 | | 2 | 1 | 2 | 1 | | | | 2 | 1 | | | | |
| 1 | | 1 | | 1 | || 1 | | 1 | | 1 | || 1 | | 1 | | 1 | || 1 | | 1 | | 1 | |
| 2 | 1 | 2 | 1 | 2 | 1 | | 2 | 1 | 2 | 1 | 2 | 1 | | 2 | 1 | 2 | 1 | 2 | 1 | | 2 | 1 | 2 | 1 | 2 | 1 |
| 1 | | 1 | | 1 | || 1 | | | | 1 | || 1 | | 1 | | | || 1 | | | | | |
考虑顺序有32种可能
|----|---|---|---|---|---|---|---|---|---|---|---------|-----|---|---|--------|----|---|--------|
| | | | | | | | | | | | 占比 | 数量 | | | | | | 实测占比 |
| 1 | | 0 | 1 | 2 | 0 | 0 | 3 | 4 | 5 | | 0.02778 | 53 | | 6 | 0.6481 | 24 | | 0.0265 |
| 2 | | 0 | 1 | 2 | 0 | 0 | 3 | 5 | 4 | | 0.02778 | 51 | | 6 | | | | 0.0255 |
| 3 | | 0 | 1 | 2 | 0 | 0 | 4 | 3 | 5 | | 0.02778 | 58 | | 6 | | | | 0.029 |
| 4 | | 0 | 1 | 2 | 0 | 0 | 5 | 3 | 4 | | 0.02778 | 60 | | 6 | | | | 0.03 |
| 5 | | 0 | 1 | 2 | 0 | 0 | 4 | 5 | 3 | | 0.02778 | 57 | | 6 | | | | 0.0285 |
| 6 | | 0 | 1 | 2 | 0 | 0 | 5 | 4 | 3 | | 0.02778 | 66 | | 6 | | | | 0.033 |
| 7 | | 0 | 1 | 0 | 0 | 4 | 2 | 3 | 0 | | 0.02778 | 58 | | 5 | 0.2685 | 7 | | 0.029 |
| 8 | | 0 | 1 | 0 | 0 | 3 | 4 | 2 | 0 | | 0.04167 | 89 | | 5 | | | | 0.0445 |
| 9 | | 0 | 1 | 0 | 0 | 4 | 3 | 2 | 0 | | 0.01389 | 20 | | 5 | | | | 0.01 |
| 10 | | 0 | 1 | 3 | 0 | 0 | 2 | 4 | 5 | | 0.04167 | 92 | | 6 | | | | 0.046 |
| 11 | | 0 | 1 | 3 | 0 | 0 | 2 | 5 | 4 | | 0.04167 | 75 | | 6 | | | | 0.0375 |
| 12 | | 0 | 1 | 4 | 0 | 0 | 2 | 3 | 5 | | 0.02778 | 57 | | 6 | | | | 0.0285 |
| 13 | | 0 | 1 | 5 | 0 | 0 | 2 | 3 | 4 | | 0.02778 | 49 | | 6 | | | | 0.0245 |
| 14 | | 0 | 1 | 3 | 0 | 0 | 4 | 2 | 5 | | 0.02083 | 40 | | 6 | | | | 0.02 |
| 15 | | 0 | 1 | 3 | 0 | 0 | 5 | 2 | 4 | | 0.02083 | 35 | | 6 | | | | 0.0175 |
| 16 | | 0 | 1 | 4 | 0 | 0 | 3 | 2 | 5 | | 0.01389 | 35 | | 6 | | | | 0.0175 |
| 17 | | 0 | 1 | 5 | 0 | 0 | 3 | 2 | 4 | | 0.01389 | 22 | | 6 | | | | 0.011 |
| 18 | | 0 | 1 | 4 | 0 | 0 | 5 | 2 | 3 | | 0.02083 | 33 | | 6 | | | | 0.0165 |
| 19 | | 0 | 1 | 5 | 0 | 0 | 4 | 2 | 3 | | 0.02083 | 37 | | 6 | | | | 0.0185 |
| 20 | | 0 | 0 | 0 | 2 | 3 | 1 | 0 | 0 | | 0.08333 | 187 | | 4 | 0.0833 | 1 | | 0.0935 |
| 21 | | 0 | 2 | 0 | 0 | 4 | 1 | 3 | 0 | | 0.02778 | 60 | | 5 | | | | 0.03 |
| 22 | | 0 | 3 | 0 | 0 | 4 | 1 | 2 | 0 | | 0.01852 | 34 | | 5 | | | | 0.017 |
| 23 | | 0 | 4 | 0 | 0 | 3 | 1 | 2 | 0 | | 0.05556 | 111 | | 5 | | | | 0.0555 |
| 24 | | 0 | 0 | 4 | 2 | 0 | 1 | 0 | 3 | | 0.08333 | 162 | | 5 | | | | 0.081 |
| 25 | | 0 | 2 | 3 | 0 | 0 | 1 | 4 | 5 | | 0.04167 | 85 | | 6 | | | | 0.0425 |
| 26 | | 0 | 2 | 3 | 0 | 0 | 1 | 5 | 4 | | 0.04167 | 74 | | 6 | | | | 0.037 |
| 27 | | 0 | 2 | 4 | 0 | 0 | 1 | 3 | 5 | | 0.02778 | 60 | | 6 | | | | 0.03 |
| 28 | | 0 | 2 | 5 | 0 | 0 | 1 | 3 | 4 | | 0.02778 | 55 | | 6 | | | | 0.0275 |
| 29 | | 0 | 3 | 4 | 0 | 0 | 1 | 2 | 5 | | 0.01852 | 28 | | 6 | | | | 0.014 |
| 30 | | 0 | 3 | 5 | 0 | 0 | 1 | 2 | 4 | | 0.01852 | 54 | | 6 | | | | 0.027 |
| 31 | | 0 | 4 | 5 | 0 | 0 | 1 | 2 | 3 | | 0.02778 | 60 | | 6 | | | | 0.03 |
| 32 | | 0 | 5 | 4 | 0 | 0 | 1 | 2 | 3 | | 0.02778 | 43 | | 6 | | | | 0.0215 |
有5个结构的有7个
|---|---|---|---|---|---|---|---|---|---------|-----|
| 0 | 1 | 0 | 0 | 4 | 2 | 3 | 0 | | 0.02778 | 58 |
| 0 | 1 | 0 | 0 | 3 | 4 | 2 | 0 | | 0.04167 | 89 |
| 0 | 1 | 0 | 0 | 4 | 3 | 2 | 0 | | 0.01389 | 20 |
| 0 | 2 | 0 | 0 | 4 | 1 | 3 | 0 | | 0.02778 | 60 |
| 0 | 3 | 0 | 0 | 4 | 1 | 2 | 0 | | 0.01852 | 34 |
| 0 | 4 | 0 | 0 | 3 | 1 | 2 | 0 | | 0.05556 | 111 |
| 0 | 0 | 4 | 2 | 0 | 1 | 0 | 3 | | 0.08333 | 162 |
分别为
|---|----|---|---|----------|---|---|-----|---|---|---------|
| 0 | 1 | | | 0.027778 | | 0 | 3 | | | 0.01852 |
| | | 4 | | | | | | 4 | | |
| 2 | 3 | | | | | 1 | 2 | | | |
| | 58 | | | | | | 34 | | | |
| 0 | 1 | | | 0.041667 | | 0 | 4 | | | 0.05556 |
| | | 3 | | | | | | 3 | | |
| 4 | 2 | | | | | 1 | 2 | | | |
| | 89 | | | | | | 111 | | | |
| 0 | 1 | | | 0.013889 | | 0 | | 4 | | 0.08333 |
| | | 4 | | | | | 2 | | | |
| 3 | 2 | | | | | 1 | | 3 | | |
| | 20 | | | | | | 162 | | | |
| 0 | 2 | | | 0.027778 | | | | | | |
| | | 4 | | | | | | | | |
| 1 | 3 | | | | | | | | | |
| | 60 | | | | | | | | | |
占比为0.26852
有4个结构的有1个
|---|---|---|---|---|---|---|---|---|---|---|-------|-----|
| 0 || 0 | 0 || 2 | 3 | 1 | 0 | 0 | | 0.083 | 187 |
| 0 | || | |||||||||
| | 2 || 3 | |||||||||
| 1 | || | |||||||||
| | 187 || | |||||||||
| | | | | | | | | | | | | |
占比为0.08333
所以平面的平均密度为（1-0.26852-0.083333）*6+0.26852*5+0.08333*4=5.56481
5.564802/6=0.92747
对比随机2000次实测占比数据和计算占比数据
同样的办法计算4*4平面的平均密度
不考虑顺序只有2种可能
|---|---|---|---|---|---|---|---|---|
| 2 | 1 | 2 | 1 | | 2 | 1 | | |
| 1 | | 1 | | | 1 | | 1 | |
| 2 | 1 | 2 | 1 | | 2 | 1 | 2 | 1 |
| 1 | | 1 | | | 1 | | | |
如果考虑顺序有5种可能
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | | 0 | 1 | | 0 | 3 | | 0 | 2 |
| | | | | | | | | | | |
| 2 | 3 | | 3 | 2 | | 1 | 2 | | 1 | 3 |
| | | | | | | | | | | |
| 0 | | | | | | | | | | |
| | 2 | | | | | | | | | |
| 1 | | | | | | | | | | |
这5种可能分成2种
|---|---|---|---|---|---|-----|
| 0 | 1 | | 0 | 1 | | |
| | | + | | | = | 0.5 |
| 2 | 3 | | 3 | 2 | | |
和
|---|---|---|---|---|---|---|---|---|-----|
| 0 | 3 | | 0 | 2 | | 0 | | | |
| | | + | | | + | | 2 | = | 0.5 |
| 1 | 2 | | 1 | 3 | | 1 | | | |
实测表明,顺序相同的情况下占比是相同的
|---|---|---|---|---|---|---------|
| 0 | 1 | 0 | 2 | 3 | | 0.25 |
| 0 | 1 | 0 | 3 | 2 | | 0.25 |
| | | | | | | |
| 0 | 3 | 0 | 1 | 2 | | 0.16667 |
| 0 | 2 | 0 | 1 | 3 | | 0.16667 |
| 0 | 0 | 2 | 1 | 0 | | 0.16667 |
(0.25+0.25)*4+(0.1667+0.1667)*4+0.1667*3=3.8333
3.8333/4= 0.9583
对比4*6的0.9275，空间变大后密度变小了。