A function is to be constructed that, given an input of a positive integer value m, automatically enumerates all possible cases. Starting with an arithmetic sequence of length 4*m + 2, two terms are removed leaving 4m terms, which are then evenly divided into m groups, each consisting of 4 numbers. It is required that the 4 numbers in each group form their own arithmetic sequence. The function should generate all combinations of the two removed terms and the corresponding m groups of four numbers forming arithmetic sequences.
To clarify the problem statement further, let's break down the requirements:
- An arithmetic sequence of length 4*m + 2 is created.
- Two terms from this sequence are removed.
- The remaining 4m terms are divided into m groups, each containing exactly 4 terms.
- Each of these groups must also be an arithmetic sequence.
- The function should output all possible combinations of the two removed terms and the resulting groups of four numbers that form arithmetic sequences.
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=JpD9cdC2.jpg&userId=1828799)
Finally, represent each outcome using the grid diagram shown above.
f[m_] := Module[{S, graph}, S = Range[4 m + 2];
graph =
RelationGraph[DisjointQ,
Select[Subsets[S, {4}], Equal @@ Differences@# &]];
GroupBy[FindClique[graph, Length /@ FindClique[graph], All],
Complement[S, Join @@ #] &]]
grid[m_] :=
Grid[MapIndexed[{#2[[1]], Multicolumn[#, 2, Spacings -> 1]} &,
Map[Function[x,
TextGrid[
Range[4 m + 2] /.
a -> Alternatives @@
Thread[a : Complement[Range[4 m + 2], Join @@ x] :>
Highlighted[a, FrameMargins -> 0]], Frame -> All,
FrameStyle -> Gray,
Background -> {None, None,
Join @@ MapIndexed[
Thread[Thread[{1, #}] ->
Lighter[Hue[(Tr@#2 - 1)/Length@x], 0.85]] &, x]}]],
Values@f[m], {2}]], Alignment -> Left, Spacings -> {Automatic, 1}]
The above code is not producing results correctly. How can it be corrected or modified to achieve the correct output?