Question

I have found conjecture to be true for all cases considered, but how do you find a deductive proof that it is true for all cases?

The functions ominvar and omplot

In[]:=
ominvar@n_:=Module[{q,r,s},​​q=Partition[Flatten@Table[{m,k},{m,n},{k,n}],2];​​r=Flatten@Table[Range@n,{n}];​​s=Partition[Flatten@Transpose@{q,r},3];​​s/.{m_Integer,u_,v_}:>τ[m,u,v]]
In[]:=
omplot@n_:=Grid[Partition[ominvar@n,n],Frame->All]
In[]:=
omplot/@Range@4
Out[]=

τ[1,1,1]
,
τ[1,1,1]
τ[1,2,2]
τ[2,1,1]
τ[2,2,2]
,
τ[1,1,1]
τ[1,2,2]
τ[1,3,3]
τ[2,1,1]
τ[2,2,2]
τ[2,3,3]
τ[3,1,1]
τ[3,2,2]
τ[3,3,3]
,
τ[1,1,1]
τ[1,2,2]
τ[1,3,3]
τ[1,4,4]
τ[2,1,1]
τ[2,2,2]
τ[2,3,3]
τ[2,4,4]
τ[3,1,1]
τ[3,2,2]
τ[3,3,3]
τ[3,4,4]
τ[4,1,1]
τ[4,2,2]
τ[4,3,3]
τ[4,4,4]


Make this a “shell game” with the “switch operator” sw

In[]:=
sw[u_,v_]@c_List:=sw[u,v]/@c
In[]:=
sw[a_,b_]@τ[u_,v_,w_]:=τ[sw[a,b]@u,sw[a,b]@v,sw[a,b]@w]
In[]:=
sw[a_Integer,b_Integer]@c_Integer:=c/.{a->b,b->a}

conjecture

Let n, u, and v be any positive integers with u and v each less than or equal to n . Then we assert
In[]:=
conjecture[n_,{u_,v_}]:=(ominvar@n)==(Sort@sw[u,v]@ominvar@n)

A test of conjecture

In[]:=
n=20;
In[]:=
Select[Partition[Flatten@Outer[List,Range@n,Range@n],2],(#[[1]]<#[[2]])&]
Out[]=
{{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{1,10},{1,11},{1,12},{1,13},{1,14},{1,15},{1,16},{1,17},{1,18},{1,19},{1,20},{2,3},{2,4},{2,5},{2,6},{2,7},{2,8},{2,9},{2,10},{2,11},{2,12},{2,13},{2,14},{2,15},{2,16},{2,17},{2,18},{2,19},{2,20},{3,4},{3,5},{3,6},{3,7},{3,8},{3,9},{3,10},{3,11},{3,12},{3,13},{3,14},{3,15},{3,16},{3,17},{3,18},{3,19},{3,20},{4,5},{4,6},{4,7},{4,8},{4,9},{4,10},{4,11},{4,12},{4,13},{4,14},{4,15},{4,16},{4,17},{4,18},{4,19},{4,20},{5,6},{5,7},{5,8},{5,9},{5,10},{5,11},{5,12},{5,13},{5,14},{5,15},{5,16},{5,17},{5,18},{5,19},{5,20},{6,7},{6,8},{6,9},{6,10},{6,11},{6,12},{6,13},{6,14},{6,15},{6,16},{6,17},{6,18},{6,19},{6,20},{7,8},{7,9},{7,10},{7,11},{7,12},{7,13},{7,14},{7,15},{7,16},{7,17},{7,18},{7,19},{7,20},{8,9},{8,10},{8,11},{8,12},{8,13},{8,14},{8,15},{8,16},{8,17},{8,18},{8,19},{8,20},{9,10},{9,11},{9,12},{9,13},{9,14},{9,15},{9,16},{9,17},{9,18},{9,19},{9,20},{10,11},{10,12},{10,13},{10,14},{10,15},{10,16},{10,17},{10,18},{10,19},{10,20},{11,12},{11,13},{11,14},{11,15},{11,16},{11,17},{11,18},{11,19},{11,20},{12,13},{12,14},{12,15},{12,16},{12,17},{12,18},{12,19},{12,20},{13,14},{13,15},{13,16},{13,17},{13,18},{13,19},{13,20},{14,15},{14,16},{14,17},{14,18},{14,19},{14,20},{15,16},{15,17},{15,18},{15,19},{15,20},{16,17},{16,18},{16,19},{16,20},{17,18},{17,19},{17,20},{18,19},{18,20},{19,20}}
In[]:=
conjecture[n,#]&/@%//Tally
Out[]=
{{True,190}}