Pole Visibility Problem Code

Arranging poles

I’m thinking about how to place 9 poles of lengths 1 to 9 parallel to the x-axis at positions 1 to 9. Each pole should face up to 3 other poles without one obstructing another's view.

Crafting the program

I’m developing a Wolfram Mathematica program to check every permutation of 9 poles to ensure lines connecting their tops intersect no other poles. This involves examining all possible pole pairings.

Ensuring visibility

When connecting poles with straight lines, only touch at the tops is allowed; intersections through poles obscure views, thus obstructing a pole's visibility.

Reassessing the idea

I'm rethinking how positioning 9 poles of varying lengths along the x-axis can ensure each pole's top points to a maximum of 3 other poles, considering their visibility.

Mapping visibility

This involves checking all pole arrangements to ensure each pole sees at most 3 others, considering their heights and positions on the x-axis, without any obstructions.

Evaluating line-of-sight

Interestingly enough, I’m ensuring that lines connecting poles don't intersect any other pole segments in their interiors, considering the height and position constraints along the x-axis.

Mapping intersections

I’m examining if any pole segments intersect along their interiors with the lines connecting pole tops. This involves checking if intersection points lie within the segments' vertical extents.

Clarifying intersection rules

OK, let me see. Lines can touch poles only at their tops, not within them. Intersections must strictly be at the poles' tops to avoid obscuring visibility.

Clarifying visibility

Poles' tops can intersect lines without issues, as they define the visible area. The bottom doesn't interfere, so endpoints are not classified as interior.

Mapping intersections

I'm piecing together how lines between pole tops might intersect, ensuring that only the top touches other poles, thereby maintaining a clear path.

Ensuring clear sight

I’m verifying that lines from pole i to j stay clear above their tops, making sure intermediate pole k meets requirements for unobstructed visibility.

Ensuring clear visibility

I'm checking that each pole's top sees up to three other poles without obstructions, using symmetry to determine unobstructed lines of sight.

Ensuring symmetry

OK, let me see. Checking each pole's visibility up to three pairs, using symmetry to verify no obstructions. Each pole's top can see three others or less.

Checking all angles

I'm working on examining every permutation of pole heights in Mathematica, ensuring each pole top sees up to three others, leveraging symmetry for obstructions.

Verifying visibility

I’m crafting Mathematica code to check if each pair of pole tops sees each other, considering obstructions by examining intermediate pole positions.

Ensuring visibility

I’m confirming each pole pair's visibility, updating counts for each pole, and checking if no pole sees more than three others. Progress is steady with these parameters in mind.

Filtering permutations

Piecing together a method to filter permutations using "Select" for efficient visibility cross-checking, aiming to pinpoint valid arrangements with optimal performance.

Confirming visibility patterns

I’m finalizing the Mathematica code to ensure each pole top sees up to three others, filtering valid permutations for efficient visibility counting. Progress is steady and promising.

Ensuring visibility

Focusing on pairs with indices where $i < j$ covers all necessary pairs, mapping out the final steps for validating permutations in Mathematica.

Finalizing the program

I’m wrapping up the Mathematica program, refining visibility checks, and ensuring performance is balanced with a large set of permutations. Adding comments for clarity is next.

Reasoned about pole arrangement problem for 22 seconds

Below is one acceptable solution in Wolfram Mathematica. In this solution we represent an “arrangement” by a permutation of

  {1, 2, 3, 4, 5, 6, 7, 8, 9}

where the pole at position i (with x‑coordinate i) has height equal to the ith entry of the permutation.

The idea is as follows. Two poles at positions i and j (with i < j) have clear line‐of‑sight (i.e. “face” each other) if for every intermediate pole at position k (with i < k < j) the top of that pole does not “stick up” into the line joining the tops of poles i and j. In formulas, if the top of pole i is at $(i, h_i)$ and that of pole j is at $(j, h_j)$, then the line joining them is

$$y = h_i + \frac{h_j-h_i}{j-i}\,(x-i).$$

So at the intermediate position $x=k$ the line has height

$$y_k = h_i+\frac{h_j-h_i}{j-i}(k-i).$$

Since the pole at k extends from $y=0$ to $y=h_k$, the line is “inside” pole k if

$$y_k < h_k.$$

(If $y_k = h_k$ the line “just touches” the top, and that is allowed.) We require that for a clear line‐of‑sight between poles i and j the condition

$$y_k \ge h_k\quad\text{for all } k \text{ with } i<k<j$$

holds.

Then we count, for each pole, how many other poles it sees (that is, for how many other poles the connecting line is unobstructed). We want every pole to “face” at most 3 others.

The complete Mathematica code is given below.

---

(* Define a function that tests whether the poles at positions i and j see each other *)
visiblePairQ[perm_List, i_Integer, j_Integer] := Module[{hi, hj},
  hi = perm[[i]];
  hj = perm[[j]];
  AllTrue[
    Range[i + 1, j - 1],
    Function[k,
      Module[{yk},
        yk = hi + (hj - hi)*(k - i)/(j - i);
        yk >= perm[[k]]
      ]
    ]
  ]
];

(* For a given permutation (arrangement), compute the number of other poles visible from each pole *)
visibleCounts[perm_List] := Module[{n, counts},
  n = Length[perm];
  counts = ConstantArray[0, n];
  Do[
    If[visiblePairQ[perm, i, j],
      counts[[i]]++;  (* pole at position i sees j *)
      counts[[j]]++;  (* and vice-versa, since the visibility is symmetric *)
    ],
    {i, 1, n - 1}, {j, i + 1, n}
  ];
  counts
];

(* Select only those arrangements where every pole sees at most 3 others *)
validArrangements = Select[
  Permutations[Range[9]],
  AllTrue[visibleCounts[#], (# <= 3) &] &
];

(* Output the number of valid arrangements found *)
Print["Number of valid arrangements: ", Length[validArrangements]];

(* Optionally, display the valid arrangements *)
validArrangements

---

Explanation

1. visiblePairQ:
   This function takes a permutation perm and two positions i and j (with i < j). It computes the height of the line joining the tops of poles at positions i and j at each intermediate position k and checks that

   yk = perm[[i]] + (perm[[j]] - perm[[i]])*(k - i)/(j - i)
   

   is at least as high as the height of the pole at k (i.e. perm[[k]]). If this is true for every such k, then the two poles see each other.

2. visibleCounts:
   This function loops over all pairs of poles (with i < j) and uses visiblePairQ to decide if the two poles are mutually visible. It then increments the “visible count” for both poles. The result is a list of 9 numbers, each giving the number of poles visible from that pole.

3. validArrangements:
   We use Select to pick only those permutations (i.e. arrangements) for which every pole’s visible count is at most 3.

4. Finally, we print out the number of valid arrangements and also (if desired) output the list of valid arrangements.

Any correct Mathematica program that checks all $9!$ arrangements and verifies that each pole sees at most 3 other poles is acceptable.