Copyright ©2014 by Paul Niquette All rights reserved. 

ere are
solutions to a collection of 11 puzzles (p =
0, 1, ...10) which were chosen to illustrating two
kinds of algorithms. Each puzzle has a sequence of
seven given numbers (i = 1, 2, ...7),
which are clues for ascertaining the Next Number.
Two solutions are shown for each puzzle. The Next
Numbers derived by the
differationale algorithm
have a blue background. The red background
indicates Next Numbers that are generated
by mathematical procedures as explained below.
Differationale
Algorithm
Here we generalize the differationale algorithm, which was applied in the solution to the Next Number puzzle, beginning with this table of symbols in the form n(i,j), constraining the values of i and j for the solutions to the puzzles in the table above...
Simple differences between successive given numbers are to be calculated and placed in column j = 1, as... n(2,1) = n(2,0)  n(1,0), n(3,1) = n(3,0)  n(2,0), ...(7,1) = n(7,0)  n(6,0).Likewise, simple differences between successive entries in column j = 1 are placed in column j = 2. Continuing à droite through the other columns (j = 3 through j = 6), we see that the last simple difference placed in column j = 6 is n(7,6). No more simple differences can be entered. In fact, since n(8,0) is unknown, none of the entries in row i = 8 can be derived by simple differences. Thus the 'differ' part of the differationale algorithm has ended. The 'rationale' part begins by making one explicit assumption: n(8,6) = n(7,6).That assumption raises an elementary question, What value for n(8,5) will make the simple difference n(8,5)  n(7,5) = n(8,6)? The answer to that elementary question is n(8,5) = n(8,6) + n(7,5). Likewise, the value of n(8,4) = n(8,5) + n(7,4), and so forth such that the Next Number solution n(8,0) = n(7,0) + n(8,1). Elementary algebra provides a shortcut: n(8,0) = n(7,0) + n(7,1) + n(7,2) + n(7,3) + n(7,4) + n(7,5) + n(7,6)Puzzle #0 has a rather weird sequence of given numbers. Mathematically, there seems to be neither rhyme nor reason for them. Not to be discouraged, our solution in the table below applies the differationale algorithm described above, along with the shortcut to derive the solution n(8,0) = 666...
The significance of those zeroes for Puzzle #0 is that apparently not all of the given numbers were 'needed' for the solution, only n(1,0) through n(4,0). The rest could be derived as successive Next Numbers. That's a logical oxymoron. Indeed, it requires the solver to make a whole raft of hidden assumptions, starting with n(5,3) = n(4,3), then n(6,3) = n(5,3),... all the way down to n(8,3) = n(7,3), then each entry gets calculated back à gauche to the j = 0 column. That makes the given numbers 183, 301, and 460 into derived numbers. Which may be appropriate for a puzzlemaster, but for a solver? Not so much. Just suppose the puzzlemaster made n(5,0) = 184 instead of 183. It is easy to show that the assumption by the solver must be n(8,6) = n(7,6) = 15 instead of 0, and the solution n(8,0) = 701 not 666.Meanwhile, the solutions for Puzzles #1 through Puzzle #10 marked in blue above were all produced exactly the same way as has been demonstrated for Puzzle #0. The solutions marked in red above are each explained beginning here... Puzzle #1seems to have an obvious mathematical solution of the form n(i,0) = 2^{(i1)} such that n(8,0) = 128. If that is what the puzzlemaster had in mind for the Next Number in Puzzle #1 then n(8,0) = 2^{7} = 128 is the solution. However, the differationale algorithm gives n(8,0) = 127 as the Next Number. Which
solution is right and which solution is
wrong?
Puzzle #2 raises the same question as Puzzle #1, with its mathematical solution of the form n(i,0) = 3^{(i1)} yielding n(8,0) = 3^{7} = 2,187 as the solution, while the differationale algorithm gives n(8,0) = 2,059 as the Next Number. Puzzles #3, #4, #5, #6 all have mathematical solutions of the form n(i,0) = i^{(p1)} and seemingly by coincidence the two solutions for each puzzle are respectively the same: 64, 512, 4,096, and 32,768. Solvers are invited to figure out why these solutions with their diverse derivations wind up agreeing with each other. Puzzle #7 makes the comparison based on factorials, n(i,0) = i!. The expression for the Next Number n(8,0) = 8! = 40,320, but by differationale algorithm, n(8,0) = 23,633. Quite a 'discrepancy'. Puzzle #8 makes the comparison based on totorials, n(i,0) = i& = i(i+1)/2. The expression for n(8,0) = 8& = 36, and by differationale algorithm, n(8,0) = 36. No 'discrepancy' at all. Puzzle #9 provides a sequence of given numbers as sums of the first i prime numbers; accordingly, n(i,0) = ∑ kprimes, for k = 0, 1, ... i, from which we obtain n(8,0) = 76, whereas by differationale algorithm, n(8,0) = 106. n(i,0) =n(i,0) =n(i,0) = Puzzle #10 provides a list of obvious numerical palindromes for which the puzzlemaster might have been expecting that the solution would be n(8,0) = 707; however, by differationale algorithm, n(8,0) = 708. Hey, 708 is not even a palindrome, so the differationale algorithm must therefore be  well, wrong. ot so fast there. Reprising a vital consideration from the puzzle page: Every puzzle is the creation of a puzzlemaster, who must formulate both the puzzle and its solution solution plus a rationalerationale for the solution. = 106(8,0) = 106 le #9 prov Puzzle #9 pro
Gratitude from the
puzzlemaster (me) to John Swanson, William LaSor,
Al Bongarzone, and Rich Alexander for their
contributions to Differationale.

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