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

by : $110$

 * An Absolute Skeleton

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