使用DuckDB SQL求三阶六角幻方

六角幻方
https://zhuanlan.zhihu.com/p/544255474
原图如下：

条件：最外圈是3个在同一边的数之和相等,还要配上对角线的5个数相等,间隔1条边的两边中点连线上的4个数和相等。
根据横排5排的和都相等，可知这个和=38（(1+19)*19/2/5）

with recursive n as(select i from range(1, 20)t(i)) --1 到 19 
, n3 as (select (1<<a)+(1<<b)m, * from(select n.i a,n1.i b, 38-n.i-n1.i c from n, n n1) where a<>b and a<>c and b<>c and c between 1 and 19 ) -- 找出和为38的三元组
,r as(select 1 lv,[a,b] s,n3.* from n3 -- 将和为38的三元组按首尾数字连成串
union all
select lv+1, s+ [n3.a,n3.b],n3.m+r.m,n3.a,n3.b,n3.c from r,n3 where r.c=n3.a and r.m&n3.m=0 and lv<6)
,t as(select * from r where lv=6 and c=s[1]) --串成闭环
,a as(select m, s,[38-s[12]-s[4],38-s[2]-s[6],38-s[4]-s[8],38-s[10]-s[6],38-s[12]-s[8],38-s[10]-s[2]]t from t) --算出每种排列中圈相邻两数和
,b as(select 1 lv,a.m+(1<<n.i)+(1<<n1.i) m,s+[n.i,n1.i] s,t,n1.i i from a,n,n n1 where a.m &(1<<n.i)=0 and a.m &(1<<n1.i)=0 and n.i+n1.i=t[1] and n.i<>n1.i -- 从剩余的数字中试拼
union all
select lv+1,m+(1<<n1.i),s+[n1.i],t,n1.i from b,n n1  where m &(1<<n1.i)=0 and n1.i+b.i=t[lv+1] and lv<5
)
select * from b where lv=5 --1 因为已知唯一解，这些都是对称、旋转后的结果
;
┌───────┬─────────┬──────────────────────────────────────────────────────────────────┬──────────────────────┬───────┐
│  lv   │    m    │                                s                                 │          t           │   i   │
│ int32 │  int64  │                             int64[]                              │       int64[]        │ int64 │
├───────┼─────────┼──────────────────────────────────────────────────────────────────┼──────────────────────┼───────┤
│     5 │ 1048542 │ [3, 19, 16, 12, 10, 13, 15, 14, 9, 11, 18, 17, 7, 2, 4, 8, 6, 1] │ [9, 6, 12, 14, 7, 8] │     1 │
│     5 │ 1048542 │ [9, 14, 15, 13, 10, 12, 16, 19, 3, 17, 18, 11, 6, 8, 4, 2, 7, 1] │ [14, 12, 6, 9, 8, 7] │     1 │
│     5 │ 1048542 │ [3, 17, 18, 11, 9, 14, 15, 13, 10, 12, 16, 19, 7, 1, 6, 8, 4, 2] │ [8, 7, 14, 12, 6, 9] │     2 │
│     5 │ 1048542 │ [10, 13, 15, 14, 9, 11, 18, 17, 3, 19, 16, 12, 4, 8, 6, 1, 7, 2] │ [12, 14, 7, 8, 9, 6] │     2 │
│     5 │ 1048542 │ [15, 14, 9, 11, 18, 17, 3, 19, 16, 12, 10, 13, 8, 6, 1, 7, 2, 4] │ [14, 7, 8, 9, 6, 12] │     4 │
│     5 │ 1048542 │ [16, 19, 3, 17, 18, 11, 9, 14, 15, 13, 10, 12, 2, 7, 1, 6, 8, 4] │ [9, 8, 7, 14, 12, 6] │     4 │
│     5 │ 1048542 │ [15, 13, 10, 12, 16, 19, 3, 17, 18, 11, 9, 14, 8, 4, 2, 7, 1, 6] │ [12, 6, 9, 8, 7, 14] │     6 │
│     5 │ 1048542 │ [18, 17, 3, 19, 16, 12, 10, 13, 15, 14, 9, 11, 1, 7, 2, 4, 8, 6] │ [8, 9, 6, 12, 14, 7] │     6 │
│     5 │ 1048542 │ [16, 12, 10, 13, 15, 14, 9, 11, 18, 17, 3, 19, 2, 4, 8, 6, 1, 7] │ [6, 12, 14, 7, 8, 9] │     7 │
│     5 │ 1048542 │ [18, 11, 9, 14, 15, 13, 10, 12, 16, 19, 3, 17, 1, 6, 8, 4, 2, 7] │ [7, 14, 12, 6, 9, 8] │     7 │
│     5 │ 1048542 │ [9, 11, 18, 17, 3, 19, 16, 12, 10, 13, 15, 14, 6, 1, 7, 2, 4, 8] │ [7, 8, 9, 6, 12, 14] │     8 │
│     5 │ 1048542 │ [10, 12, 16, 19, 3, 17, 18, 11, 9, 14, 15, 13, 4, 2, 7, 1, 6, 8] │ [6, 9, 8, 7, 14, 12] │     8 │
├───────┴─────────┴──────────────────────────────────────────────────────────────────┴──────────────────────┴───────┤
│ 12 rows                                                                                                 5 columns │
└───────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘
Run Time (s): real 0.242 user 0.265625 sys 0.125000