Searching for certain triples in a list - wolfram-mathematica

Question

Let’s assume we have a list of elements of the type {x,y,z} for x, y
and z integers. And, if needed x < y < z.
We also assume that the list contains at least 3 such triples.
Can Mathematica easily solve the following problem?
To detect at least one triple of the type {a,b,.}, {b,c,.} and {a,c,.}?
I am more intereseted in an elegant 1-liner than computational efficient solutions.

Answer 1

If I understood the problem, you want to detect triples not necessarily following one another, but generally present somewhere in the list. Here is one way to detect all such triples. First, some test list:
In[71]:= tst = RandomInteger[5,{10,3}]
Out[71]= {{1,1,0},{1,3,5},{3,3,4},{1,2,1},{2,0,3},{2,5,1},{4,2,2},
{4,3,4},{1,4,2},{4,4,3}}
Here is the code:
In[73]:=
Apply[Join,ReplaceList[tst,{___,#1,___,#2,___,#3,___}:>{fst,sec,th}]&###
Permutations[{fst:{a_,b_,_},sec:{b_,c_,_},th:{a_,c_,_}}]]
Out[73]= {{{1,4,2},{4,3,4},{1,3,5}},{{1,4,2},{4,2,2},{1,2,1}}}
This may perhaps satisfy your "one-liner" requirement, but is not very efficient. If you need only triples following one another, then, as an alternative to solution given by #Chris, you can do
ReplaceList[list,
{___, seq : PatternSequence[{a_, b_, _}, {b_, c_, _}, {a_,c_, _}], ___} :> {seq}]

Answer 2

I don't know if I interpreted your question correctly but suppose your list is something like
list = Sort /# RandomInteger[10, {20, 3}]
(*
{{3, 9, 9}, {0, 5, 6}, {3, 4, 8}, {4, 6, 10}, {3, 6, 9}, {1, 4, 8},
{0, 6, 10}, {2, 9, 10}, {3, 5, 9}, {6, 7, 9}, {0, 9, 10}, {1, 7, 10},
{4, 5, 10}, {0, 2, 5}, {0, 6, 7}, {1, 8, 10}, {1, 8, 10}}
*)
then you could do something like
ReplaceList[Sort[list],
{___, p:{a_, b_, _}, ___, q:{a_, c_, _}, ___, r:{b_, c_, _}, ___} :> {p, q, r}]
(* Output:
{{{0, 2, 5}, {0, 9, 10}, {2, 9, 10}}, {{3, 4, 8}, {3, 5, 9},
{4, 5, 10}}, {{3, 4, 8}, {3, 6, 9}, {4, 6, 10}}}
*)
Note that this works since it is given that for any element {x,y,z} in the original list we have x<=y. Therefore, for a triple {{a,b,_}, {a,c,_}, {b,c,_}} \[Subset] list we know that a<=b<=c. This means that the three elements {a,b,_}, {a,c,_}, and {b,c,_} will appear in that order in Sort[list].

Answer 3

To match triples "of the type {a,b,.}, {b,c,.} and {a,c,.}":
list = {{34, 37, 8}, {74, 32, 65}, {48, 77, 18}, {77, 100, 30},
{48, 100, 13}, {100, 94, 55}, {48, 94, 73}, {77, 28, 12},
{90, 91, 51}, {34, 5, 32}};
Cases[Partition[list, 3, 1], {{a_, b_, _}, {b_, c_, _}, {a_, c_, _}}]

Answer 4

(Edited)
(Tuples was not the way to go)
Do you require something like:
list = RandomInteger[10, {50, 3}];
Cases[Permutations[
list, {3}], {{a_, b_, _}, {b_, c_, _}, {a_, c_, _}} /; a < b < c]
giving
{{{0, 1, 2}, {1, 5, 2}, {0, 5, 4}},
{{2, 3, 5},{3, 4, 10}, {2, 4, 5}},
{{6, 8, 10}, {8, 10, 10},{6, 10, 0}},
{{2, 4, 5}, {4, 8, 2}, {2, 8, 5}},
{{2, 4, 5}, {4, 7, 7}, {2, 7, 3}},
{{0, 2, 2}, {2, 7, 3}, {0, 7, 2}},
{{0, 2, 1}, {2, 7, 3}, {0, 7, 2}}}
or perhaps (as other have interpreted the question):
Cases[Permutations[
list, {3}], {{a_, b_, _}, {b_, c_, _}, {a_, c_, _}}];