The puzzle shown above has eight solutions. I was amazed to discover that in a sequence of 500 puzzles, one puzzle had 190 solutions, and fourteen puzzles had exactly one solution. Since I started from the "answer," there are no puzzles generated that will have no solution.

The sequence of integers in Table 2 is the sorted list of the numbers of solutions found in each of the 500 trials.
 

 Table 3 is a compressed frequency table. The left-hand column lists the number of puzzles (out of 500) that had the number of solutions listed in the middle column.  Thus, 21 puzzles had three solutions, 17 puzzles had nine solutions and 17 puzzles had 10 solutions, etc. The right-hand column lists the cumulative number of puzzles accounted for. Thus, the cumulative total of 58 at the end of the third row indicates the number of puzzles that had 3, 6, or 8 solutions. The cumulative total of 92 at the end of the fourth row indicates 34 more puzzles have been counted, 17 with 9 solutions, and 17 with 10 solutions.
 

Figure 1 provides a discrete histogram of the data in Table 1.

 I don't know the conditions on the original 4 × 4 array of integers in the interval  that account for the number of solutions. I suspect they might have something to do with the number of partitions of the row, column, and diagonal sums. Indeed, if such a sum were, say, 25, then there would be

partitions using integers in the interval . The first of these partitions, namely,

suggests why the number of partitions is so high. We need to restrict the partitions to just those containing four integers. This much shorter list is then

Note that each of these 23 partitions is sorted, from smallest to largest integer. These integers might appear permuted in the solution to a Challenger puzzle in which 25 were a row, column, or diagonal sum. The combinatorics of this approach do not look fruitful.

The code I used to generate a puzzle appears in Table 4.

 
The eight lists of four pairs at the top of the table represent valid configurations for the four numbers to be revealed. I noticed that the puzzles in my local newspaper had exactly one such number in each row, column, and diagonal. (However, on the internet, I found puzzle generators that did not adhere to this restriction.) The eight lists in Table 4 are exhaustive - they represent the only ways to satisfy the constraint stated.

One such configuration is selected at random. Then, , the full 4 × 4 array of integers in the interval , is obtained randomly, and a matrix  is constructed. This matrix contains the four displayed digits, and the remaining twelve entries are the variables .  From , the row, column, and diagonal sums can be computed, and from , these sums can be expressed as ten equations in the twelve unknowns .

Maple's solve and isolve (integer-solve) commands provide the same solution, and for the cases I examined, each solution contained not just two indeterminates, but three. A typical such solution (vector) appears in Table 5, where we have called the three indeterminates .

 Each component of this vector must be an integer in the interval , a condition that generates 24 linear inequalities.  Maple does solve this set of inequalities, but it takes a noticeable amount of time, and the solutions themselves can contain further inequalities. For these reasons, a "brute-force" solution was adopted for vectors like the one in Table 5. All possible integers in the range  were substituted for each of , , and ; then the twelve components of the resulting vector were examined for the maximum and minimum.  If the maximum was no larger than 9, and the minimum no smaller than 1, the solution was accepted as valid. In fact, to test the validity of such solutions, the row, column, and diagonal sums were checked and found to be correct.

I imagine that a puzzle with nearly 200 solutions would be more difficult to solve that one with just one or two solutions. That would probably be because most empty squares would admit so many different integers. Table 6 contains a puzzle with some 185 solutions. The rightmost column displays, for each cell in the 4 × 4 grid, the integers that appear in at least one of the 185 solutions.

At first, I thought the blue cells in Table 6 were a surprise. These cells contain the displayed integers for the puzzle, and in every one of the 185 solutions, these cells contain just the single displayed integer. However, the uncolored cells are the ones corresponding to the variables , so it is only in these uncolored cells that multiple integers can appear.

The complete code for creating, displaying, and solving these Challenger puzzles is available behind the "New Puzzle" button, and the Code Edit Region icon.  Simply bring up the Context Menu on these components; for the "New Puzzle" button, select "Component Properties" and click on the Edit button; for the Code Edit Region, select "Expand Code Edit Region."