Missing Numbers
Copyright 2014 by Paul Niquette. All rights reserved.
Missing Numbers

In the Differationale puzzle, solvers gasped upon reading this extraordinary passage...


As an algorithm, differationale will be found to have immense power.  It provides a solution to any next-number puzzle throughout the known universe and in perpetuity.



...which asserts an infinity kind of thing as it would pertain to a particular puzzle kind of thing -- any puzzle that postulates a sequence of given numbers and asks for one number -- the next number in the series.  The Differationale puzzle demonstrated how the algorithm [a] takes into account all of the given numbers without regard to how they were created and [b] derives a solution, which may or may not comply with what the puzzle-master had in mind when creating the puzzle, but [c] guarantees one and only one solution -- complete with a rationale

An unstated assumption is that the given numbers and the next number form a contiguous series, which is a feature exemplified by the Next Number puzzle and  thousands of challenges in the On-Line Encyclopedia of Integer Sequences (OEIS).

Here you are invited to consider the reverse of that challenge. In a Missing Numbers puzzle, only the first and last numbers in a series are given, and the solution comprises all the numbers in between.  Solvers of the King for a Day puzzle will recognize that kind of thing...
 
Steps:
?
?
?
?
?
?
?
1
?
?
?
?
?
?
666

...for which a linear interpolation algorithm would apply the fact that (666 - 1)/7 = 95...
 
Steps:

+95


+95


+95

+95
+95
+95
+95
1
96
191
286
381
476
571
666
...but we should note here that one and only one such solution will result from linear interpolation.

Nota bene, if the last number were, say, 667 instead of 666, (667 - 1)/7 = 95.14285~ we get...

Steps:

+95.14285~


+95.14285~


+95.14285~

+95.14285~
+95.14285~
+95.14285~
+95.14285~
1
95.14285~
191.286~
286.429~
381.571~
476.714~
571.857~
667
 
...which would be an OK kind of thing if the puzzle-master allows for non-integer solutions.

C onventional challenges authorize only integers for number sequences.  Accordingly, linear interpolation will not always 'work'.  Solvers will be able to show that a reverse version of the differationale algorithm will always 'work'.  Let's call it the Missing Numbers algorithm. 
Hey, and the Missing Numbers algorithm is not limited to one and only one solution.  If you would like to figure out how that is possible, here is a question for you...



How many solutions to this
M
issing Numbers

puzzle can you find?

 1
?
?
100



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