Missing Numbers

Copyright ©2014 by Paul Niquette All rights reserved.

 1
?
?
100

How many solutions to this Missing Numbers puzzle can you find?  For starters, since (100 - 1)/3 = 33, linear interpolation will 'work': n(i+1) = n(i) + 33 for i = 1, 2, 3, giving you integers n(2) = 34 and n(3) = 67 and an answer to the question, "One single, solitary solution."  Oh, but you may find other formulas that 'work'... 
For example, n(i) = 100(i-1)/3 for i = 1, 2, 3, 4, but that gives you a solution in which n(2) = 4.641589~ and n(3) = 21.54435~.  Again your answer to the question is, "One single, solitary solution." 
Not only that but those numbers are not integers.  The formula n(i) = 100(i-1)/3 gives you irrational numbers, which will be acceptable but only if the puzzle-master allows for non-integer solutions.  Do not despair.  Soon, you will be able to answer the question in the puzzle this way...

 

"The missing numbers algorithm is able to produce a quantity of solutions greater than total number of subatomic particles in the Milky Way galaxy."

 

Here is the algorithm from the solution page of the Differationale puzzle modified to fit the Missing Numbers puzzle...


j = 0

j = 1

j = 2

i = 1

n1

 

 

i = 2

n2

n2-n1

 

i = 3

n3

n3-n2

(n3-n2) - (n2-n1)

i = 4

n4




In the j = 0 column, we positioned unknown numbers n2 and n3 in between the two given numbers n1 and n4.  Simple differences between successive numbers are inserted in the table as indicated in columns j = 1 and j = 2.  Based on the differationale algorithm, the following expression for n4 can be written by summing all the terms in the i = 3 row:
n4 = n3 + (n3 - n2) + (n3 - n2) - (n2 - n1) and rearranged to read thus...
n4 - n1 = 3(n3 - n2)
Let us define n4 - n1 = 3(n3 - n2) as the characteristic formula for the missing numbers algorithm as applied to this puzzle, which has a total of four numbers in the series.  The left member of the equation is the simple difference between the first number in the series n1 and the last number n4, which will be called the span of the given numbers.  The right member of the characteristic formula 3(n3 - n2) is an expression mandating how the span must be distributed among the unknown numbers to meet the requirements of the puzzle. 

For an integer solution, the characteristic formula shows that in this case the span must be divisible by three. Meanwhile, the unknown numbers can be regarded as solver-chosen variables, which are related to one another as constrained by the given numbers.  
There may be other constraints, beginning with the requirement for integer solutions, as already mentioned.  Unmentioned is an apparent requirement that solutions be positive numbers.  Then too, sophisticated solvers may have noticed that only monotonic solutions have been shown so far, wherein n(i+1) > n(i).
Whatever the constraints imposed upon the unknowns, it will be a handy idea to make use of the characteristic formula to relate them as variables in functions of one another, thus the missing numbers algorithm provides these relationships...
n3 = n2 + (n4 - n1)/3 or
n2 = n3 - (n4 - n1)/3
...which must hold for any solution to this puzzle to be valid.  Go ahead and experiment with this puzzle.  First you can substitute the two given numbers in the puzzle, 1 and 100...
n3 = n2 + (100 - 1)/3 = n2 + 33.
...and then you can chose any number for n2 and merely set n3 equal to that same number plus 33 and you will have produced missing numbers as specified in the puzzle.  How many? 
Suppose that, on behalf of an unidentified puzzle-master, you impose the following constraints on the solution to the Missing Numbers puzzle: positive monotonically increasing integers only. 
To comply with all those constraints n2 > n1, and the smallest such value is n2 = 2.  Meanwhile n3 < n4, and the largest such value is n3 = 99.  But n2 = n3 - 33, so the largest value for n2 = 66.  Accordingly, n2 = 2, 3, 4,...66, giving you the following answer to the question:

 

65 solutions

 

All 65 solutions produced by the Missing Numbers algorithm are depicted in this scatter chart, showing the relationship between the solver-assigned values for n2 and the requisite values for n3 that will assure compliance with the indicated requirements...


...however, if you systematically remove the assumed constraints imposed by the unidentified puzzle-master you will obtain a whole raft of solutions by the Missing Numbers algorithm as represent in this tabulation...

TypSMPI

MinSMPI

MaxSMPI

TypSPI

TypSI

TypSRatn

TypSIrN

1

1

1

1

1

1

1

34

2

66

80

-20

1.42857143

3.14159265

67

35

99

113

13

34.4285714

36.1415927

100

100

100

100

100

100

100

Rational

Integers

Positive


Monotonic




 106(8,0) = 106
le #9 prov
Puzzle #9 pro
Solutions by Missing Numbers Algorithm



TypSMPI Typical Sequence of Monotonic Positive Integers (includes Linear Interpolation)

MinSMPI Miniimum Sequence of Monotonic Positive Integers

MaxSMPI Maximum Sequence of Monotonic Positive Integers

TypSPI Typical Sequence of Positive Integers (includes non-monotonic n3 > n4)

TypSI Typical Sequence of Integers (includes negative n2 < 0)


TypSRatn Typical Sequence of Rational Numbers (includes n2 = 10/7)

TypSIrN Typical Sequence of Irrational Numbers (includes n2 = pi)

For sequences having a 'span' that includes only two unknowns, you have been able to apply elementary mathematical expressions to obtain an unboundable collection of solutions using the missing numbers algorithm.  The process for larger spans can be illustrated using the challenge in the King for a Day puzzle...

1
?
?
?
?
?
?
666

...wherein you see the two given numbers, n1 = 1 and n8 = 666.  Now, in between, you have six unknowns to determine n2, n3,... n7 in rendering a solution.  That could require you to do a whole lot of guessing.  Bummer.

Solvers who calculate the 'span' of the series, 666 - 1 = 665, will surely take notice that 665/7 = 95, thus you have...

Steps 

1

 

3

 

3

 

4

 

5

 

6

 

7

 

 

+95

 

+95

 

+95

 

+95

 

+95

 

+95

 

+95

 

1

 

96

 

191

 

286

 

381

 

476

 

571

 

666

n1

 

n2

 

n3

 

n4

 

n5

 

n6

 

n7

 

n8


...which is one solution for sure, and it's based on linear interpolation. 

Using the differationale algorithm, you can derive the characteristic formula for the missing numbers algorithm...
7[(n7 - n2) - 3(n6 - n3) + 5(n5 - n4)] = n8 - n1
...wherein you see that for integer solutions, the span must be divisible by seven, which has already been confirmed.

The six unknowns for this puzzle have been related to one another in pairs of differences.  To simplify our finding of solutions, let us give each pair a name: x = n7 - n2, y = n6 - n3, and z = n5 - n4.   Three variables are a whole lot fewer to make guesses about than six.  Well, duh.  But that is only part of your solution strategy.  The characteristic formula can be rewritten...
x - 3y + 5z = (n8 - n1)/7
You already have one solution for the puzzle -- the solution derived by linear interpolation --  which is perfectly valid.  You might regard it as a base case.  For the puzzle at hand you can calculate as follows:
x = 571 - 96 = 475, y = 476 - 191 = 285, z = 381 - 286 = 95 and confirm that 475 - 3(285) + 5(95) = 95
Flexing any number of solutions off the base case, you are able to write...
n7 = n2 + 475, n6 = n3 + 285, and n5 = n4+ 95
...and observe how n7 is completely determined by n2, n6 by n3, n5 by n4.  Get out your spreadsheet and plug in any old values for n2, n3, n4 and watch what happens to n7, n6, n5.  You will be able to produce an unboundable quantity of solutions in every numerical category allowed by the puzzle-master except one -- sequences of monotonic positive integers, for which the total quantity is in fact bounded...

TypSMPI

MinSMPI

MaxSMPI

TypSPI

TypSI

TypSRatn

TypSIrN

1

1

1

1

1

1

1

96

2

190

571

-10

1.42857~

2.71828~

191

3

379

476

-20

3.33333~

3.14159~

286

4

568

381

0

34.42857~

31.6227~

381

99

663

476

95

124.42857~

126.6227~

476

288

664

761

265

288.33333~

288.1416~

571

477

665

1,046

465

476.42857~

477.71828~

666

666

666

666

666

666

666

Rational

Integers

Positive


Monotonic




Nota Bene, even with its immense power, your missing numbers algorithm does not produce all possible solutions, only those that can be flexed from the base case that you used (which is reprised in the TypSMPI column).  Indeed, you will not find, for example, that weird series in the King for a Day puzzle here.  But that's an OK kind of thing.  All you have to do is go back and re-derive your characteristic formula based on that series, and off you go. There are unboundably more solutions to be found by the missing numbers algorithm.   Each base case fosters a 'family' of solutions.
Finally, let us address the question in the puzzle, which calls for the quantity of solutions.  Except for sequences of monotonic positive integers, the quantity of solutions will be unboundable.  But you may want to focus on those crummy old boundable sequences.  If so, you can make use of the two columns labelled MinSMPI and MaxSMPI, where you will see three relevant ranges...
1< n2 < 191, 2 < n3 < 380, 3 < n4 < 569  respectively making 476 < n7 < 666, 287 < n6 < 665, 98 < n5 < 664
Inasmuch as n2, n3, n4 are independent of one another, their respective ranges can be multiplied, such that...
188 x 376 x 565 = 39,868,032
...which is the quantity of solutions made available by the missing numbers algorithm for that SMPI 'family'.

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