First, let us review the original puzzle. The limerick provided the clues shown below. The solution was straight forward using the method described in the Next Number puzzle. In the format of these tables, the subtractions of adjacent numbers produce the entries immediately higher up. The sophisticated solver assumes that the top-most difference -- 6 in this case -- prevails for the whole table. He or she then performs indicated additions to produce the entries immediately below until the requisite numbers are revealed in the bottom-most row.
 

he puzzle derived from the king's limerick removes three of the given numbers in the original puzzle (15, 46, 100).  As shown in the table below only two numbers are given for the series of eight numbers: the first, which is one ("one star") and the last which is 666 ("Mark of Beast").  The objective is to fill in six missing numbers ("six paces free") with no clues for how they might be derived. 

One might suppose that the king assumed that there was only one solution and it was elementary indeed: the king's tormentor would notice immediately that (666 - 1)/7 = 95, thus by merely adding 95 to entries in the series for each step, one readily fills in the missing numbers...

Thus the king's tormentor would receive his reward and rule the Land of Puzzles for just one day, and that would be that. But one might also suppose that the king's tormentor had something else in mind: To rule the Land of Puzzles for more than one day.  The 'linear' solution above is only one of a kind.  Whereas any old numbers might be inserted in the series, a vital question must answered: "What is the rationale for those numbers?  All of them.

n order to produce the solution for the series in king's limerick, we would seem to need huge amounts of trial-and-error, inasmuch as there are six unknowns. While enjoying his prize as King for a Day, the king's tormentor would not have much time to do a whole lot of guessing about numbers in order to extend his rule over the Land of Puzzles.  There is a way to reduce the amount of guessing, by working the method described above backwards. 

First, assume some value (10 instead of 6, say)  for the common difference and fill in the top-most row of the table. Starting at the right-most column, the sophisticated solver sees that the top and bottom entries are already determined, leaving only two differences to be inserted into the table. He or she makes a guess (31 and 138, say) for each of them, then calculates backwards filling in the table entries until the left-most entry is determined. Almost certainly, it will not have the required value of one. Having a computer with a spreadsheet program sure helps.

The case indicated below used a common difference of 10 and, through trial-and-error, the other two numbers were re-computed a few times until that one appeared as the first number in the series. Here's how the process would look...

Start with the known solution to the original limerick. That provides a basis for choosing those two numbers in the right-most column (31 and 138). After all, we have a pair that we know will produce a one over in the left-most position, specifically 47 and 206. With some experimenting, it turns out that another combination which works is 45 and 200.

A whole family of solutions can be generated by simply adding and subtracting 2 and 6 to the respective differences in the right-most column and working the table from right to left to produce all the intervening numbers.

h well, since 2014 there has been an alternative. Solvers of the Missing Numbers puzzle will acquire a powerful tool for solving all such puzzles that might ever come along. The table below gives a few selected solutions.

1	1	1	1	1	1
15	5	10	20	40	80
46	10	20	40	80	160
100	15	30	80	120	320
183	98	113	163	203	403
301	265	275	295	335	415
460	450	455	465	485	525
666	666	666	666	666	666