Henry Ernest Dudeney/Puzzles and Curious Problems/111 - Odds and Evens/Solution/Declarations

Puzzles and Curious Problems by Henry Ernest Dudeney: $111$

Odds and Evens

Every asterisk and letter represents a figure,

and "$\mathtt O$" stands for an odd figure ($1$, $3$, $5$, $7$, or $9$)

while "$\mathtt E$" represents an even figure ($2$, $4$, $6$, $8$, or $0$).

Can you construct an arrangement complying with these conditions?

There are $6$ solutions.

Can you find one, or all of them?

       *******
    ----------
 ***)*********
     OE*
     -----
      ****
      OO**
      -----
        ***
        EE*
        ----
         ***
         EO*
         ----
         ****
         EE**
         -----
           ***
           OO*
           ---

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 $q_7$ denote the digits of $Q$ which are calculated at each stage of the long division process in turn.

Let $n_1$ to $n_7$ denote the partial dividends which are subject to the $1$st to $7$th division operations respectively.

Note that $n_2 = 0$, but has been retained for consistency of numbering.

Let $j_1$ to $j_7$ denote the least significant digits of $n_1$ to $n_7$ as they are brought down from $N$ at each stage of the long division process in turn.

Let $p_1$ to $p_7$ denote the partial products generated by the $1$st to $7$th division operations respectively: $p_k = q_k D$

Again note that $p_2 = 0$, but has been retained for consistency of numbering.

Let $d_1$ to $d_7$ 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
     OE*        --> p_1
     -----
      ****      --> n_3
      OO**      --> p_3
      -----
        ***     --> n_4
        EE*     --> p_4
        ----
         ***    --> n_5
         EO*    --> p_5
         ----
         ****   --> n_6
         EE**   --> p_6
         -----
           ***  --> n_7
           OO*  --> p_7
           ---