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

by : $111$

 * Odds and Evens

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 ---