How to make a list of three distinct points are not collinear so that their coordinates consist of nine different numbers from 1 to 9?

Question

I am trying to make a list of three distinct points are not collinear so that their coordinates consist of nine different numbers from 1 to 9,

{{1, 9, 8}, {5, 2, 3}, {7, 4, 6}}

I tried

tab = Table[{{a, b, c}, {x, y, z}, {m, n, t}}, {a, 9}, {b, 9}, {c, 
    9}, {x, 9}, {y, 9}, {z, 9}, {m, 9}, {n, 9}, {t, 9}];

SystemException["MemoryAllocationFailure"]

I tried again

tab = Table[{{a, b, c}, {x, y, z}, {m, n, t}}, {a, 3}, {b, 3}, {c, 3}, {x, 3, 6}, {y, 3, 6}, {z, 3, 6}, {m, 7, 9}, {n, 7, 9}, {t, 7, 9}];
ss = Flatten[tab, 8];
result = Pick[ss, Not@*CollinearPoints /@ ss];
list = Select[result, (9 == Length[Union @@ #] &)]

{{{1, 2, 3}, {4, 5, 6}, {7, 9, 8}}, {{1, 2, 3}, {4, 5, 6}, {8, 7, 9}}, {{1, 2, 3}, {4, 5, 6}, {8, 9, 7}}, {{1, 2, 3}, {4, 5, 6}, {9, 7, 8}}, {{1, 2, 3}, {4, 5, 6}, {9, 8, 7}}, {{1, 2, 3}, {4, 6, 5}, {7, 8, 9}}, {{1, 2, 3}, {4, 6, 5}, {7, 9, 8}}, {{1, 2, 3}, {4, 6, 5}, {8, 7, 9}}, {{1, 2, 3}, {4, 6, 5}, {8, 9, 7}}, {{1, 2, 3}, {4, 6, 5}, {9, 7, 8}}, {{1, 2, 3}, {4, 6, 5}, {9, 8, 7}}, {{1, 2, 3}, {5, 4, 6}, {7, 8, 9}}, {{1, 2, 3}, {5, 4, 6}, {7, 9, 8}}, {{1, 2, 3}, {5, 4, 6}, {8, 7, 9}}, {{1, 2, 3}, {5, 4, 6}, {8, 9, 7}}, {{1, 2, 3}, {5, 4, 6}, {9, 7, 8}}, {{1, 2, 3}, {5, 4, 6}, {9, 8, 7}}, {{1, 2, 3}, {5, 6, 4}, {7, 8, 9}}, {{1, 2, 3}, {5, 6, 4}, {7, 9, 8}}, {{1, 2, 3}, {5, 6, 4}, {8, 7, 9}}, {{1, 2, 3}, {5, 6, 4}, {8, 9, 7}}, {{1, 2, 3}, {5, 6, 4}, {9, 7, 8}}, {{1, 2, 3}, {5, 6, 4}, {9, 8, 7}}, {{1, 2, 3}, {6, 4, 5}, {7, 8, 9}}, {{1, 2, 3}, {6, 4, 5}, {7, 9, 8}}, {{1, 2, 3}, {6, 4, 5}, {8, 7, 9}}, {{1, 2, 3}, {6, 4, 5}, {8, 9, 7}}, {{1, 2, 3}, {6, 4, 5}, {9, 7, 8}}, {{1, 2, 3}, {6, 4, 5}, {9, 8, 7}}, {{1, 2, 3}, {6, 5, 4}, {7, 8, 9}}, {{1, 2, 3}, {6, 5, 4}, {7, 9, 8}}, {{1, 2, 3}, {6, 5, 4}, {8, 7, 9}}, {{1, 2, 3}, {6, 5, 4}, {8, 9, 7}}, {{1, 2, 3}, {6, 5, 4}, {9, 7, 8}}, {{1, 2, 3}, {6, 5, 4}, {9, 8, 7}}, {{1, 3, 2}, {4, 5, 6}, {7, 8, 9}}, {{1, 3, 2}, {4, 5, 6}, {7, 9, 8}}, {{1, 3, 2}, {4, 5, 6}, {8, 7, 9}}, {{1, 3, 2}, {4, 5, 6}, {8, 9, 7}}, {{1, 3, 2}, {4, 5, 6}, {9, 7, 8}}, {{1, 3, 2}, {4, 5, 6}, {9, 8, 7}}, {{1, 3, 2}, {4, 6, 5}, {7, 8, 9}}, {{1, 3, 2}, {4, 6, 5}, {8, 7, 9}}, {{1, 3, 2}, {4, 6, 5}, {8, 9, 7}}, {{1, 3, 2}, {4, 6, 5}, {9, 7, 8}}, {{1, 3, 2}, {4, 6, 5}, {9, 8, 7}}, {{1, 3, 2}, {5, 4, 6}, {7, 8, 9}}, {{1, 3, 2}, {5, 4, 6}, {7, 9, 8}}, {{1, 3, 2}, {5, 4, 6}, {8, 7, 9}}, {{1, 3, 2}, {5, 4, 6}, {8, 9, 7}}, {{1, 3, 2}, {5, 4, 6}, {9, 7, 8}}, {{1, 3, 2}, {5, 4, 6}, {9, 8, 7}}, {{1, 3, 2}, {5, 6, 4}, {7, 8, 9}}, {{1, 3, 2}, {5, 6, 4}, {7, 9, 8}}, {{1, 3, 2}, {5, 6, 4}, {8, 7, 9}}, {{1, 3, 2}, {5, 6, 4}, {8, 9, 7}}, {{1, 3, 2}, {5, 6, 4}, {9, 7, 8}}, {{1, 3, 2}, {5, 6, 4}, {9, 8, 7}}, {{1, 3, 2}, {6, 4, 5}, {7, 8, 9}}, {{1, 3, 2}, {6, 4, 5}, {7, 9, 8}}, {{1, 3, 2}, {6, 4, 5}, {8, 7, 9}}, {{1, 3, 2}, {6, 4, 5}, {8, 9, 7}}, {{1, 3, 2}, {6, 4, 5}, {9, 7, 8}}, {{1, 3, 2}, {6, 4, 5}, {9, 8, 7}}, {{1, 3, 2}, {6, 5, 4}, {7, 8, 9}}, {{1, 3, 2}, {6, 5, 4}, {7, 9, 8}}, {{1, 3, 2}, {6, 5, 4}, {8, 7, 9}}, {{1, 3, 2}, {6, 5, 4}, {8, 9, 7}}, {{1, 3, 2}, {6, 5, 4}, {9, 7, 8}}, {{1, 3, 2}, {6, 5, 4}, {9, 8, 7}}, {{2, 1, 3}, {4, 5, 6}, {7, 8, 9}}, {{2, 1, 3}, {4, 5, 6}, {7, 9, 8}}, {{2, 1, 3}, {4, 5, 6}, {8, 7, 9}}, {{2, 1, 3}, {4, 5, 6}, {8, 9, 7}}, {{2, 1, 3}, {4, 5, 6}, {9, 7, 8}}, {{2, 1, 3}, {4, 5, 6}, {9, 8, 7}}, {{2, 1, 3}, {4, 6, 5}, {7, 8, 9}}, {{2, 1, 3}, {4, 6, 5}, {7, 9, 8}}, {{2, 1, 3}, {4, 6, 5}, {8, 7, 9}}, {{2, 1, 3}, {4, 6, 5}, {8, 9, 7}}, {{2, 1, 3}, {4, 6, 5}, {9, 7, 8}}, {{2, 1, 3}, {4, 6, 5}, {9, 8, 7}}, {{2, 1, 3}, {5, 4, 6}, {7, 8, 9}}, {{2, 1, 3}, {5, 4, 6}, {7, 9, 8}}, {{2, 1, 3}, {5, 4, 6}, {8, 9, 7}}, {{2, 1, 3}, {5, 4, 6}, {9, 7, 8}}, {{2, 1, 3}, {5, 4, 6}, {9, 8, 7}}, {{2, 1, 3}, {5, 6, 4}, {7, 8, 9}}, {{2, 1, 3}, {5, 6, 4}, {7, 9, 8}}, {{2, 1, 3}, {5, 6, 4}, {8, 7, 9}}, {{2, 1, 3}, {5, 6, 4}, {8, 9, 7}}, {{2, 1, 3}, {5, 6, 4}, {9, 7, 8}}, {{2, 1, 3}, {5, 6, 4}, {9, 8, 7}}, {{2, 1, 3}, {6, 4, 5}, {7, 8, 9}}, {{2, 1, 3}, {6, 4, 5}, {7, 9, 8}}, {{2, 1, 3}, {6, 4, 5}, {8, 7, 9}}, {{2, 1, 3}, {6, 4, 5}, {8, 9, 7}}, {{2, 1, 3}, {6, 4, 5}, {9, 7, 8}}, {{2, 1, 3}, {6, 4, 5}, {9, 8, 7}}, {{2, 1, 3}, {6, 5, 4}, {7, 8, 9}}, {{2, 1, 3}, {6, 5, 4}, {7, 9, 8}}, {{2, 1, 3}, {6, 5, 4}, {8, 7, 9}}, {{2, 1, 3}, {6, 5, 4}, {8, 9, 7}}, {{2, 1, 3}, {6, 5, 4}, {9, 7, 8}}, {{2, 1, 3}, {6, 5, 4}, {9, 8, 7}}, {{2, 3, 1}, {4, 5, 6}, {7, 8, 9}}, {{2, 3, 1}, {4, 5, 6}, {7, 9, 8}}, {{2, 3, 1}, {4, 5, 6}, {8, 7, 9}}, {{2, 3, 1}, {4, 5, 6}, {8, 9, 7}}, {{2, 3, 1}, {4, 5, 6}, {9, 7, 8}}, {{2, 3, 1}, {4, 5, 6}, {9, 8, 7}}, {{2, 3, 1}, {4, 6, 5}, {7, 8, 9}}, {{2, 3, 1}, {4, 6, 5}, {7, 9, 8}}, {{2, 3, 1}, {4, 6, 5}, {8, 7, 9}}, {{2, 3, 1}, {4, 6, 5}, {8, 9, 7}}, {{2, 3, 1}, {4, 6, 5}, {9, 7, 8}}, {{2, 3, 1}, {4, 6, 5}, {9, 8, 7}}, {{2, 3, 1}, {5, 4, 6}, {7, 8, 9}}, {{2, 3, 1}, {5, 4, 6}, {7, 9, 8}}, {{2, 3, 1}, {5, 4, 6}, {8, 7, 9}}, {{2, 3, 1}, {5, 4, 6}, {8, 9, 7}}, {{2, 3, 1}, {5, 4, 6}, {9, 7, 8}}, {{2, 3, 1}, {5, 4, 6}, {9, 8, 7}}, {{2, 3, 1}, {5, 6, 4}, {7, 8, 9}}, {{2, 3, 1}, {5, 6, 4}, {7, 9, 8}}, {{2, 3, 1}, {5, 6, 4}, {8, 7, 9}}, {{2, 3, 1}, {5, 6, 4}, {9, 7, 8}}, {{2, 3, 1}, {5, 6, 4}, {9, 8, 7}}, {{2, 3, 1}, {6, 4, 5}, {7, 8, 9}}, {{2, 3, 1}, {6, 4, 5}, {7, 9, 8}}, {{2, 3, 1}, {6, 4, 5}, {8, 7, 9}}, {{2, 3, 1}, {6, 4, 5}, {8, 9, 7}}, {{2, 3, 1}, {6, 4, 5}, {9, 7, 8}}, {{2, 3, 1}, {6, 4, 5}, {9, 8, 7}}, {{2, 3, 1}, {6, 5, 4}, {7, 8, 9}}, {{2, 3, 1}, {6, 5, 4}, {7, 9, 8}}, {{2, 3, 1}, {6, 5, 4}, {8, 7, 9}}, {{2, 3, 1}, {6, 5, 4}, {8, 9, 7}}, {{2, 3, 1}, {6, 5, 4}, {9, 7, 8}}, {{2, 3, 1}, {6, 5, 4}, {9, 8, 7}}, {{3, 1, 2}, {4, 5, 6}, {7, 8, 9}}, {{3, 1, 2}, {4, 5, 6}, {7, 9, 8}}, {{3, 1, 2}, {4, 5, 6}, {8, 7, 9}}, {{3, 1, 2}, {4, 5, 6}, {8, 9, 7}}, {{3, 1, 2}, {4, 5, 6}, {9, 7, 8}}, {{3, 1, 2}, {4, 5, 6}, {9, 8, 7}}, {{3, 1, 2}, {4, 6, 5}, {7, 8, 9}}, {{3, 1, 2}, {4, 6, 5}, {7, 9, 8}}, {{3, 1, 2}, {4, 6, 5}, {8, 7, 9}}, {{3, 1, 2}, {4, 6, 5}, {8, 9, 7}}, {{3, 1, 2}, {4, 6, 5}, {9, 7, 8}}, {{3, 1, 2}, {4, 6, 5}, {9, 8, 7}}, {{3, 1, 2}, {5, 4, 6}, {7, 8, 9}}, {{3, 1, 2}, {5, 4, 6}, {7, 9, 8}}, {{3, 1, 2}, {5, 4, 6}, {8, 7, 9}}, {{3, 1, 2}, {5, 4, 6}, {8, 9, 7}}, {{3, 1, 2}, {5, 4, 6}, {9, 7, 8}}, {{3, 1, 2}, {5, 4, 6}, {9, 8, 7}}, {{3, 1, 2}, {5, 6, 4}, {7, 8, 9}}, {{3, 1, 2}, {5, 6, 4}, {7, 9, 8}}, {{3, 1, 2}, {5, 6, 4}, {8, 7, 9}}, {{3, 1, 2}, {5, 6, 4}, {8, 9, 7}}, {{3, 1, 2}, {5, 6, 4}, {9, 7, 8}}, {{3, 1, 2}, {5, 6, 4}, {9, 8, 7}}, {{3, 1, 2}, {6, 4, 5}, {7, 8, 9}}, {{3, 1, 2}, {6, 4, 5}, {7, 9, 8}}, {{3, 1, 2}, {6, 4, 5}, {8, 7, 9}}, {{3, 1, 2}, {6, 4, 5}, {8, 9, 7}}, {{3, 1, 2}, {6, 4, 5}, {9, 8, 7}}, {{3, 1, 2}, {6, 5, 4}, {7, 8, 9}}, {{3, 1, 2}, {6, 5, 4}, {7, 9, 8}}, {{3, 1, 2}, {6, 5, 4}, {8, 7, 9}}, {{3, 1, 2}, {6, 5, 4}, {8, 9, 7}}, {{3, 1, 2}, {6, 5, 4}, {9, 7, 8}}, {{3, 1, 2}, {6, 5, 4}, {9, 8, 7}}, {{3, 2, 1}, {4, 5, 6}, {7, 8, 9}}, {{3, 2, 1}, {4, 5, 6}, {7, 9, 8}}, {{3, 2, 1}, {4, 5, 6}, {8, 7, 9}}, {{3, 2, 1}, {4, 5, 6}, {8, 9, 7}}, {{3, 2, 1}, {4, 5, 6}, {9, 7, 8}}, {{3, 2, 1}, {4, 5, 6}, {9, 8, 7}}, {{3, 2, 1}, {4, 6, 5}, {7, 8, 9}}, {{3, 2, 1}, {4, 6, 5}, {7, 9, 8}}, {{3, 2, 1}, {4, 6, 5}, {8, 7, 9}}, {{3, 2, 1}, {4, 6, 5}, {8, 9, 7}}, {{3, 2, 1}, {4, 6, 5}, {9, 7, 8}}, {{3, 2, 1}, {4, 6, 5}, {9, 8, 7}}, {{3, 2, 1}, {5, 4, 6}, {7, 8, 9}}, {{3, 2, 1}, {5, 4, 6}, {7, 9, 8}}, {{3, 2, 1}, {5, 4, 6}, {8, 7, 9}}, {{3, 2, 1}, {5, 4, 6}, {8, 9, 7}}, {{3, 2, 1}, {5, 4, 6}, {9, 7, 8}}, {{3, 2, 1}, {5, 4, 6}, {9, 8, 7}}, {{3, 2, 1}, {5, 6, 4}, {7, 8, 9}}, {{3, 2, 1}, {5, 6, 4}, {7, 9, 8}}, {{3, 2, 1}, {5, 6, 4}, {8, 7, 9}}, {{3, 2, 1}, {5, 6, 4}, {8, 9, 7}}, {{3, 2, 1}, {5, 6, 4}, {9, 7, 8}}, {{3, 2, 1}, {5, 6, 4}, {9, 8, 7}}, {{3, 2, 1}, {6, 4, 5}, {7, 8, 9}}, {{3, 2, 1}, {6, 4, 5}, {7, 9, 8}}, {{3, 2, 1}, {6, 4, 5}, {8, 7, 9}}, {{3, 2, 1}, {6, 4, 5}, {8, 9, 7}}, {{3, 2, 1}, {6, 4, 5}, {9, 7, 8}}, {{3, 2, 1}, {6, 4, 5}, {9, 8, 7}}, {{3, 2, 1}, {6, 5, 4}, {7, 8, 9}}, {{3, 2, 1}, {6, 5, 4}, {7, 9, 8}}, {{3, 2, 1}, {6, 5, 4}, {8, 7, 9}}, {{3, 2, 1}, {6, 5, 4}, {8, 9, 7}}, {{3, 2, 1}, {6, 5, 4}, {9, 7, 8}}}

I limit $a$ from 1 to 3, $b$ from 4 to 6 and $c$ from 7 to 9.

score 6 · Accepted Answer · answered Jan 05 '24 at 01:14

$Version

(* "13.3.1 for Mac OS X ARM (64-bit) (July 24, 2023)" *)

Clear["Global`*"]

perm = Permutations[Range[9]];

Length@perm

(* 362880 *)

sel = Select[Partition[#, 3] & /@ perm, ! CollinearPoints[#] &];

Length@sel

(* 361944 *)

Since the order of the points does not matter this can be reduced to 1/6 of the sel

sel2 = DeleteDuplicates[Sort /@ sel];
Length@sel2
(* 60324 *)

For example,

sel2[[1 ;; 9]]
(* {{{1, 2, 3}, {4, 5, 6}, {7, 9, 8}}, {{1, 2, 3}, {4, 5, 6}, {8, 7, 9}}, {{1, 2,
    3}, {4, 5, 6}, {8, 9, 7}}, {{1, 2, 3}, {4, 5, 6}, {9, 7, 8}}, {{1, 2, 
   3}, {4, 5, 6}, {9, 8, 7}}, {{1, 2, 3}, {4, 5, 7}, {6, 8, 9}}, {{1, 2, 
   3}, {4, 5, 7}, {6, 9, 8}}, {{1, 2, 3}, {4, 5, 7}, {8, 6, 9}}, {{1, 2, 
   3}, {4, 5, 7}, {8, 9, 6}}} *)
Area /@ Triangle /@ %
(* {3 Sqrt[3/2], 3 Sqrt[3/2], 9/Sqrt[2], 9/Sqrt[2], 
 3 Sqrt[6], 7/2, Sqrt[115/2], Sqrt[185]/2, 19/Sqrt[2]} *)

Can I use CoplanarPoints to find the equation of the planes containing the points in the above sel2? — John Paul Peter, Jan 05 '24 at 05:02
Do not ask new questions in comments. Post a new question if necessary; however, look at Equation of the plane passing through the three points — Bob Hanlon, Jan 05 '24 at 05:58

How to make a list of three distinct points are not collinear so that their coordinates consist of nine different numbers from 1 to 9?

1 Answers1