如果我们在游泳池中有数字1到22的彩票抽奖。我们必须画出四个数字。 4个数字组合的总数为7,315。
现在是困难的部分，假设您只想保留4个中不超过2个的组合。
例如，你绘制组合1,2,3,4然后1,2,3,5或2,3,4,6是不好的。下一个有效组合是1,2,5,6，因为没有三个数字与1,2,3,4相同。
如何从1,2,3,4到19,20,21,22中取出所有其他组合中只有两个相同数字的组合。
或者找出有多少有效组合的公式是什么？
以下是前几个有效组合。 （1,2,3,4）:( -1,2,5,6-）:( 1,2,7,8-）:( 1,2,9,10）:( 1,2,11,12）： （1,2,13,14）...
感谢您提前提供任何帮助， DJ
答案 0 :(得分：1)
这是一种相当不优雅的蛮力方法。
#!/usr/bin/python
from sets import Set
setlist = []
# Generate all sets
# Sets are guaranteed to have only unique numbers
# Note: Python ranges are inclusive on the left but exclusive on the right
for first in range(1,23):
for second in range(first+1,23):
for third in range(second+1,23):
for fourth in range(third+1,23):
possible = Set([first, second, third, fourth])
# Only take sets of 4 digits
if len(possible) == 4:
setlist.append(possible)
# Print the total number of combinations
print len(setlist)
answers = []
for aset in setlist:
# Append the first answer automatically
if len(answers) == 0:
answers.append(aset)
ok = True
#Check against all our previous answers
for answer in answers:
# We have more than two intersecting values, this is not an answer
if (len(answer.intersection(aset)) > 2):
ok = False
break;
# If our answer is ok
if ok:
answers.append(aset)
# Clean up our answers and sort them all
clean = []
for answer in answers:
temp = list(answer)
temp.sort()
clean.append(temp)
# Print the clean answers
print clean
# Print the total
print len(answers)
答案 1 :(得分：0)
您正在寻找的是一种组合块设计。基本集中的元素数量为22，而每个块中的元素数量为4.此外，块必须具有不会在多个块中出现三元组（例如，{1,2,3}）的属性，这是一种更精确的说法，“你只想保留4种中只有2种是相同的组合”。
要获得块数的上限，首先考虑单个块（例如，{1,2,3,4}）。在该区块中有四个三元组：{1,2,3}，{1,2,4}，{1,3,4}和{2,3,4}。 22个元素的三元组总数 22 C 3 = 22x21x20 /（3x2x1）= 1540.由于每个块包含4个三元组，因此最大块数为1540 / 4 = 385.这并不是说有可能找到385个区块，但我们马上就会看到。
请注意，正如另一张海报所尝试的那样，通过“贪婪”方法生成设计几乎肯定注定要失败。贪婪算法早期的选择将对以后的选择数量产生负面影响，这可能会导致在找到385个区块之前陷入僵局。
如果你发现一个块设计，其中每个三元组发生完全一次，而不是最多一次，你找到了一个3-（22,4,1）块设计。 3是三元组，22是基本元素的数量，4是块的大小，1是每个三元组出现的次数。这种块设计也称为Steiner四重系统。
在Steiner四重系统上已经完成了一些学术工作。特别是Hanani，H。“On Quadruple Systems。”加拿达。 J. Math。 12,145-157,1960显示，当且仅当元素的数量与2或4 mod 6一致时，四元系统是可能的。由于22与4 mod 6一致，所以在你的情况下有一个四重系统，所以可以找到一整套385块。
Hanani的论文也提供了一个明确的结构，这在这里描述得太复杂了，但幸运的是在FOSS Sage系统中已经实现了。您可以免费下载Sage，只需运行print designs.steiner_quadruple_system(22)即可获得385块的列表。请注意，编号从0开始，因此如果您希望数字从1到22，只需为每个数字添加1，如果您想要一个具有除{1,2,3,4}以外的连续四个数字的块，您可以询问对于重新编号（或者只是通过向设计中的每个数字添加k mod 22来自己动手）。
以下是Sage提供的块：
blocks=[[0, 1, 2, 3], [0, 1, 4, 5],
[0, 1, 6, 21], [0, 1, 7, 8], [0, 1, 9, 17], [0, 1, 10, 16], [0, 1, 11,
19], [0, 1, 12, 18], [0, 1, 13, 20], [0, 1, 14, 15], [0, 2, 4, 6], [0,
2, 5, 21], [0, 2, 7, 9], [0, 2, 8, 17], [0, 2, 10, 15], [0, 2, 11, 20],
[0, 2, 12, 19], [0, 2, 13, 18], [0, 2, 14, 16], [0, 3, 4, 21], [0, 3, 5,
6], [0, 3, 7, 10], [0, 3, 8, 16], [0, 3, 9, 15], [0, 3, 11, 18], [0, 3,
12, 20], [0, 3, 13, 19], [0, 3, 14, 17], [0, 4, 7, 11], [0, 4, 8, 19],
[0, 4, 9, 20], [0, 4, 10, 17], [0, 4, 12, 15], [0, 4, 13, 16], [0, 4,
14, 18], [0, 5, 7, 12], [0, 5, 8, 18], [0, 5, 9, 16], [0, 5, 10, 20],
[0, 5, 11, 15], [0, 5, 13, 17], [0, 5, 14, 19], [0, 6, 7, 13], [0, 6, 8,
15], [0, 6, 9, 18], [0, 6, 10, 19], [0, 6, 11, 16], [0, 6, 12, 17], [0,
6, 14, 20], [0, 7, 14, 21], [0, 7, 15, 20], [0, 7, 16, 19], [0, 7, 17,
18], [0, 8, 9, 10], [0, 8, 11, 12], [0, 8, 13, 14], [0, 8, 20, 21], [0,
9, 11, 13], [0, 9, 12, 14], [0, 9, 19, 21], [0, 10, 11, 14], [0, 10, 12,
13], [0, 10, 18, 21], [0, 11, 17, 21], [0, 12, 16, 21], [0, 13, 15, 21],
[0, 15, 16, 17], [0, 15, 18, 19], [0, 16, 18, 20], [0, 17, 19, 20], [1,
2, 4, 21], [1, 2, 5, 6], [1, 2, 7, 17], [1, 2, 8, 9], [1, 2, 10, 14],
[1, 2, 11, 18], [1, 2, 12, 20], [1, 2, 13, 19], [1, 2, 15, 16], [1, 3,
4, 6], [1, 3, 5, 21], [1, 3, 7, 16], [1, 3, 8, 10], [1, 3, 9, 14], [1,
3, 11, 20], [1, 3, 12, 19], [1, 3, 13, 18], [1, 3, 15, 17], [1, 4, 7,
19], [1, 4, 8, 11], [1, 4, 9, 16], [1, 4, 10, 20], [1, 4, 12, 14], [1,
4, 13, 17], [1, 4, 15, 18], [1, 5, 7, 18], [1, 5, 8, 12], [1, 5, 9, 20],
[1, 5, 10, 17], [1, 5, 11, 14], [1, 5, 13, 16], [1, 5, 15, 19], [1, 6,
7, 14], [1, 6, 8, 13], [1, 6, 9, 19], [1, 6, 10, 18], [1, 6, 11, 17],
[1, 6, 12, 16], [1, 6, 15, 20], [1, 7, 9, 10], [1, 7, 11, 12], [1, 7,
13, 15], [1, 7, 20, 21], [1, 8, 14, 20], [1, 8, 15, 21], [1, 8, 16, 18],
[1, 8, 17, 19], [1, 9, 11, 15], [1, 9, 12, 13], [1, 9, 18, 21], [1, 10,
11, 13], [1, 10, 12, 15], [1, 10, 19, 21], [1, 11, 16, 21], [1, 12, 17,
21], [1, 13, 14, 21], [1, 14, 16, 17], [1, 14, 18, 19], [1, 16, 19, 20],
[1, 17, 18, 20], [2, 3, 4, 5], [2, 3, 6, 21], [2, 3, 7, 15], [2, 3, 8,
14], [2, 3, 9, 10], [2, 3, 11, 19], [2, 3, 12, 18], [2, 3, 13, 20], [2,
3, 16, 17], [2, 4, 7, 20], [2, 4, 8, 15], [2, 4, 9, 11], [2, 4, 10, 19],
[2, 4, 12, 17], [2, 4, 13, 14], [2, 4, 16, 18], [2, 5, 7, 14], [2, 5, 8,
20], [2, 5, 9, 12], [2, 5, 10, 18], [2, 5, 11, 17], [2, 5, 13, 15], [2,
5, 16, 19], [2, 6, 7, 18], [2, 6, 8, 19], [2, 6, 9, 13], [2, 6, 10, 17],
[2, 6, 11, 14], [2, 6, 12, 15], [2, 6, 16, 20], [2, 7, 8, 10], [2, 7,
11, 13], [2, 7, 12, 16], [2, 7, 19, 21], [2, 8, 11, 16], [2, 8, 12, 13],
[2, 8, 18, 21], [2, 9, 14, 19], [2, 9, 15, 18], [2, 9, 16, 21], [2, 9,
17, 20], [2, 10, 11, 12], [2, 10, 13, 16], [2, 10, 20, 21], [2, 11, 15,
21], [2, 12, 14, 21], [2, 13, 17, 21], [2, 14, 15, 17], [2, 14, 18, 20],
[2, 15, 19, 20], [2, 17, 18, 19], [3, 4, 7, 14], [3, 4, 8, 20], [3, 4,
9, 19], [3, 4, 10, 11], [3, 4, 12, 16], [3, 4, 13, 15], [3, 4, 17, 18],
[3, 5, 7, 20], [3, 5, 8, 15], [3, 5, 9, 18], [3, 5, 10, 12], [3, 5, 11,
16], [3, 5, 13, 14], [3, 5, 17, 19], [3, 6, 7, 19], [3, 6, 8, 18], [3,
6, 9, 16], [3, 6, 10, 13], [3, 6, 11, 15], [3, 6, 12, 14], [3, 6, 17,
20], [3, 7, 8, 9], [3, 7, 11, 17], [3, 7, 12, 13], [3, 7, 18, 21], [3,
8, 11, 13], [3, 8, 12, 17], [3, 8, 19, 21], [3, 9, 11, 12], [3, 9, 13,
17], [3, 9, 20, 21], [3, 10, 14, 18], [3, 10, 15, 19], [3, 10, 16, 20],
[3, 10, 17, 21], [3, 11, 14, 21], [3, 12, 15, 21], [3, 13, 16, 21], [3,
14, 15, 16], [3, 14, 19, 20], [3, 15, 18, 20], [3, 16, 18, 19], [4, 5,
6, 21], [4, 5, 7, 15], [4, 5, 8, 14], [4, 5, 9, 17], [4, 5, 10, 16], [4,
5, 11, 12], [4, 5, 13, 20], [4, 5, 18, 19], [4, 6, 7, 16], [4, 6, 8,
17], [4, 6, 9, 14], [4, 6, 10, 15], [4, 6, 11, 13], [4, 6, 12, 19], [4,
6, 18, 20], [4, 7, 8, 12], [4, 7, 9, 13], [4, 7, 10, 18], [4, 7, 17,
21], [4, 8, 9, 18], [4, 8, 10, 13], [4, 8, 16, 21], [4, 9, 10, 12], [4,
9, 15, 21], [4, 10, 14, 21], [4, 11, 14, 17], [4, 11, 15, 16], [4, 11,
18, 21], [4, 11, 19, 20], [4, 12, 13, 18], [4, 12, 20, 21], [4, 13, 19,
21], [4, 14, 15, 19], [4, 14, 16, 20], [4, 15, 17, 20], [4, 16, 17, 19],
[5, 6, 7, 17], [5, 6, 8, 16], [5, 6, 9, 15], [5, 6, 10, 14], [5, 6, 11,
18], [5, 6, 12, 13], [5, 6, 19, 20], [5, 7, 8, 11], [5, 7, 9, 19], [5,
7, 10, 13], [5, 7, 16, 21], [5, 8, 9, 13], [5, 8, 10, 19], [5, 8, 17,
21], [5, 9, 10, 11], [5, 9, 14, 21], [5, 10, 15, 21], [5, 11, 13, 19],
[5, 11, 20, 21], [5, 12, 14, 16], [5, 12, 15, 17], [5, 12, 18, 20], [5,
12, 19, 21], [5, 13, 18, 21], [5, 14, 15, 18], [5, 14, 17, 20], [5, 15,
16, 20], [5, 16, 17, 18], [6, 7, 8, 20], [6, 7, 9, 11], [6, 7, 10, 12],
[6, 7, 15, 21], [6, 8, 9, 12], [6, 8, 10, 11], [6, 8, 14, 21], [6, 9,
10, 20], [6, 9, 17, 21], [6, 10, 16, 21], [6, 11, 12, 20], [6, 11, 19,
21], [6, 12, 18, 21], [6, 13, 14, 15], [6, 13, 16, 17], [6, 13, 18, 19],
[6, 13, 20, 21], [6, 14, 16, 18], [6, 14, 17, 19], [6, 15, 16, 19], [6,
15, 17, 18], [7, 8, 13, 21], [7, 8, 14, 15], [7, 8, 16, 17], [7, 8, 18,
19], [7, 9, 12, 21], [7, 9, 14, 16], [7, 9, 15, 17], [7, 9, 18, 20], [7,
10, 11, 21], [7, 10, 14, 17], [7, 10, 15, 16], [7, 10, 19, 20], [7, 11,
14, 18], [7, 11, 15, 19], [7, 11, 16, 20], [7, 12, 14, 19], [7, 12, 15,
18], [7, 12, 17, 20], [7, 13, 14, 20], [7, 13, 16, 18], [7, 13, 17, 19],
[8, 9, 11, 21], [8, 9, 14, 17], [8, 9, 15, 16], [8, 9, 19, 20], [8, 10,
12, 21], [8, 10, 14, 16], [8, 10, 15, 17], [8, 10, 18, 20], [8, 11, 14,
19], [8, 11, 15, 18], [8, 11, 17, 20], [8, 12, 14, 18], [8, 12, 15, 19],
[8, 12, 16, 20], [8, 13, 15, 20], [8, 13, 16, 19], [8, 13, 17, 18], [9,
10, 13, 21], [9, 10, 14, 15], [9, 10, 16, 17], [9, 10, 18, 19], [9, 11,
14, 20], [9, 11, 16, 18], [9, 11, 17, 19], [9, 12, 15, 20], [9, 12, 16,
19], [9, 12, 17, 18], [9, 13, 14, 18], [9, 13, 15, 19], [9, 13, 16, 20],
[10, 11, 15, 20], [10, 11, 16, 19], [10, 11, 17, 18], [10, 12, 14, 20],
[10, 12, 16, 18], [10, 12, 17, 19], [10, 13, 14, 19], [10, 13, 15, 18],
[10, 13, 17, 20], [11, 12, 13, 21], [11, 12, 14, 15], [11, 12, 16, 17],
[11, 12, 18, 19], [11, 13, 14, 16], [11, 13, 15, 17], [11, 13, 18, 20],
[12, 13, 14, 17], [12, 13, 15, 16], [12, 13, 19, 20], [14, 15, 20, 21],
[14, 16, 19, 21], [14, 17, 18, 21], [15, 16, 18, 21], [15, 17, 19, 21],
[16, 17, 20, 21], [18, 19, 20, 21]]