Differationale

Copyright ©2014 by Paul Niquette All rights reserved.
Here 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. 

Puzzle:

p = 0

p = 1

p = 2

p = 3

p = 4

p = 5

p = 6

p = 7

p = 8

p = 9

p = 10

First

i = 1

1

1

1

1

1

1

1

1

1

1

1

 

i = 2

15

2

3

4

8

16

32

2

3

4

101

 

i = 3

46

4

9

9

27

81

243

6

6

9

202

 

i = 4

100

8

27

16

64

256

1,024

24

10

16

303

 

i = 5

183

16

81

25

125

625

3,125

120

15

27

404

 

i = 6

301

32

243

36

216

1,296

7,776

720

21

48

505

Last

i = 7

460

64

729

49

343

2,401

16,807

5,040

28

57

606

Next

i = 8

666

127

2,059

64

512

4,096

32,768

23,633

36

106

708

Next

i = 8

 N/A

128

2,187

64

512

4,096

32,768

40,320

36

76

707
"N/A" means "not applicable" -- there is no known solution by any mathematical procedure apart from the differationale algorithm.

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...

Given

Solution by Differationale Algorithm

j = 0

j = 1

j = 2

j = 3

j = 4

j = 5

j = 6

i = 1

n(1,0)

 

 

 

 

 

 

i = 2

n(2,0)

n(2,1)

 

 

 

 

 

i = 3

n(3,0)

n(3,1)

n(3,2)

 

 

 

 

i = 4

n(4,0)

n(4,1)

n(4,2)

n(4,3)

 

 

 

i = 5

n(5,0)

n(5,1)

n(5,2)

n(5,3)

n(5,4)

 

 

i = 6

n(6,0)

n(6,1)

n(6,2)

n(6,3)

n(6,4)

n(6,5)

 

i = 7

n(7,0)

n(7,1)

n(7,2)

n(7,3)

n(7,4)

n(7,5)

n(7,6)

i = 8

n(8,0)

n(8,1)

n(8,2)

n(8,3)

n(8,4)

n(8,5)

n(8,6)

All eleven puzzles have seven given numbers, n(1,0) through n(7,0), and the unknown Next Number will be derived as n(8,0). 

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...

Given

Solution by Differationale Algorithm

j = 0

j = 1

j = 2

j = 3

j = 4

j = 5

j = 6

i = 1

1

N.A.

 

 

 

 

 

i = 2

15

14

 

 

 

 

 

i = 3

46

31

17

 

 

 

 

i = 4

100

54

23

6

 

 

 

i = 5

183

83

29

6

0

 

 

i = 6

301

118

35

6

0

0

 

i = 7

460

159

41

6

0

0

0

i = 8

666

= 460 + 159 + 41+ 6 + 0 + 0 + 0

Of course, solvers will observe that the entries in columns j = 4, j =5, and j = 6 are all zero, which is a consequence of the fact that the simple differences in column j = 3 are all the same, n(4,3) = n(4,3) = ... n(7,3) = 6.  Truth be known, the given numbers in the puzzle were 'cooked up' by the puzzle-master for Puzzle #0, and those common differences in column j = 3 were treated as a topic in the solution to the Next Number puzzle. 

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 puzzle-master, but for a solver?  Not so much.
Just suppose the puzzle-master 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(i-1) such that n(8,0) = 128.  If that is what the puzzle-master had in mind for the Next Number in Puzzle #1 then  n(8,0) = 27 = 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(i-1) yielding n(8,0) = 37 = 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(p-1) 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) = ∑ k-primes, 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 puzzle-master 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. 

Not so fast there.  Reprising a vital consideration from the puzzle page: Every puzzle is the creation of a puzzle-master, 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

 

Writing on behalf of the puzzle-master (me), the right solutions to all eleven puzzles above are mandated to be those provided by the differationale algorithm. 

All other solutions, whatever their rationales, are therefore wrong.

 

Gratitude from the puzzle-master (me) to John Swanson, William LaSor, Al Bongarzone, and Rich Alexander for their contributions to Differationale.

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