Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution/Declarations

Puzzles and Curious Problems by Henry Ernest Dudeney: $110$

An Absolute Skeleton

Here is a good skeleton puzzle.

The only conditions are:

$(1)$ No digit appears twice in any row of figures except the dividend.

$(2)$ If $2$ be added to the last figure in the quotient it equals the last but one,

and if $2$ be added to the third figure from the end it gives the last figure but $3$ in the quotient.

That is to say, the quotient might end in, say, $9742$, or in $3186$.

        ********
    ------------
 ***)***********
      ***
       ---
       ***
       ***
       ----
       ****
       ****
       -----
         ***
         ***
         ----
         ****
         ****
         -----
          ****
          ****
          -----
           ****
           ****
           -----
            ****
            ****
            ----

Declarations

This section declares the variables which are to be used during the deduction of the solution to this skeleton puzzle.

Let $D$ denote the divisor.

Let $Q$ denote the quotient.

Let $N$ denote the dividend.

Let $q_1$ to $q_8$ denote the digits of $Q$ which are calculated at each stage of the long division process in turn.

Let $n_1$ to $n_8$ denote the partial dividends which are subject to the $1$st to $8$th division operations respectively.

Let $j_1$ to $j_8$ denote the least significant digits of $n_1$ to $n_8$ as they are brought down from $N$ at each stage of the long division process in turn.

Let $p_1$ to $p_8$ denote the partial products generated by the $1$st to $8$th division operations respectively: $p_k = q_k D$

Let $d_1$ to $d_8$ denote the differences between the partial dividends and partial products: $d_k = n_k - p_k$.

By the mechanics of a long division, we have throughout that:

$n_k = 10 d_{k - 1} + j_k$

for $k \ge 2$.

Hence we can refer to elements of the structure of this long division as follows:

        ********  -->     Q
    ------------        ---
 ***)***********  --> D ) N
      ***         --> p_1
       ---
       ***        --> n_2
       ***        --> p_2
       ----
       ****       --> n_3
       ****       --> p_3
       -----
         ***      --> n_4
         ***      --> p_4
         ----
         ****     --> n_5
         ****     --> p_5
         -----
          ****    --> n_6
          ****    --> p_6
          -----
           ****   --> n_7
           ****   --> p_7
           -----
            ****  --> n_8
            ****  --> p_8
            ----