Copyright ©1997 by Paul Niquette. All
rights reserved. 
owever firm and
fair his reputation, the King of the Land of Puzzles may
not have been especially sophisticated. For he must have
thought that the limerick
which stumped his Council of Advisors was so difficult
that there could not possibly be another solution.
The king's tormentor knew better.
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 topmost 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
bottommost 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 trialanderror,
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 topmost row of the table. Starting at the rightmost 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 leftmost 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 trialanderror,
the other two numbers were recomputed a few times
until that 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 rightmost column (31 and 138). After all, we have a pair that we know will produce a over in the leftmost 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 rightmost 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.
