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

by : $111$

 * Odds and Evens

Declarations
First we note that the partial products are structured as follows:
 * $p_1: \mathtt {O E} *$
 * $p_3: \mathtt {O O} * *$
 * $p_4: \mathtt {E E} *$
 * $p_5: \mathtt {E O} *$
 * $p_6: \mathtt {E E} * *$
 * $p_7: \mathtt {O O} *$

and so are all distinct.

Hence $q_1$ to $q_7$ are also distinct.

We also know that $q_2 = 0$ as $2$ digits were pulled down into $n_3$.

Thus the first $3$ digits of $n_3$ form a number strictly smaller than $D$.

We have that $p_1$, $p_4$, $p_5$ and $p_7$ each have $3$ digits.

But each of $q_1$, $q_4$, $q_5$ and $q_7$ are distinct.

Hence:

Notice that $p_6$ is a $4$-digit number that starts with an even number.

Therefore $p_6 \ge 2000$.

Hence:

That is:
 * $223 \le D \le 249$

The four $3$-digit multiples of $D$ are of the patterns:
 * $p_1: \mathtt {O E} *$, $p_4: \mathtt {E E} *$, $p_5: \mathtt {E O} *$, $p_7: \mathtt {O O} *$

If $3 D < 700$, the first three multiples of $D$ all begin with the digits $2$, $4$ and $6$, which causes the candidate $D$ to fail the criteria above.

Thus:
 * $3 D \ge 700$

or equivalently:
 * $D \ge 234$

We check the rest of candidates for $D$ to see whether they have multiples that are of each of the six forms.

First we check the $3$-digit multiples.

For visual clarity, the first two digits are marked $\color {red} {\text {red} }$ when odd and $\color {blue} {\text {blue} }$ when even:


 * $\begin{array} {r|rrrrrrrr} \times & 2 & 3 & 4 \\ \hline

\color {blue} 2 \color {red} 3 4 & \color {blue} 4 \color {blue} 68 & \color {red} 7 \color {blue} 02 & \color {red} 9 \color {red} 3 6 & \\ \color {blue} 2 \color {red} 3 5 & \color {blue} 4 \color {red} 70 & \color {red} 7 \color {blue} 05 & \color {red} 9 \color {blue} 40 & \\ \color {blue} 2 \color {red} 3 6 & \color {blue} 4 \color {red} 72 & \color {red} 7 \color {blue} 08 & \color {red} 9 \color {blue} 44 & \\ \color {blue} 2 \color {red} 3 7 & \color {blue} 4 \color {red} 74 & \color {red} 7 \color {red} 11 & \color {red} 9 \color {blue} 48 & \\ \color {blue} 2 \color {red} 3 8 & \color {blue} 4 \color {red} 76 & \color {red} 7 \color {red} 14 & \color {red} 9 \color {red} 52 & \\ \color {blue} 2 \color {red} 3 9 & \color {blue} 4 \color {red} 78 & \color {red} 7 \color {red} 17 & \color {red} 9 \color {red} 56 & \\ \color {blue} 2 \color {blue} 40 & \color {blue} 4 \color {blue} 80 & \color {red} 7 \color {blue} 20 &  \color {red} 9 \color {blue} 60 & \\ \color {blue} 2 \color {blue} 41 & \color {blue} 4 \color {blue} 82 & \color {red} 7 \color {blue} 23 &  \color {red} 9 \color {blue} 64 & \\ \color {blue} 2 \color {blue} 42 & \color {blue} 4 \color {blue} 84 & \color {red} 7 \color {blue} 26 &  \color {red} 9 \color {blue} 68 & \\ \color {blue} 2 \color {blue} 43 & \color {blue} 4 \color {blue} 86 & \color {red} 7 \color {blue} 29 &  \color {red} 9 \color {red} 72 & \\ \color {blue} 2 \color {blue} 44 & \color {blue} 4 \color {blue} 88 & \color {red} 7 \color {red} 32 &  \color {red} 9 \color {red} 76 & \\ \color {blue} 2 \color {blue} 45 & \color {blue} 4 \color {red} 90 & \color {red} 7 \color {red} 35 &  \color {red} 9 \color {blue} 80 & \\ \color {blue} 2 \color {blue} 46 & \color {blue} 4 \color {red} 92 & \color {red} 7 \color {red} 38 &  \color {red} 9 \color {blue} 84 & \\ \color {blue} 2 \color {blue} 47 & \color {blue} 4 \color {red} 94 & \color {red} 7 \color {blue} 41 &  \color {red} 9 \color {blue} 88 & \\ \color {blue} 2 \color {blue} 48 & \color {blue} 4 \color {red} 96 & \color {red} 7 \color {blue} 44 &  \color {red} 9 \color {red} 92 & \\ \color {blue} 2 \color {blue} 49 & \color {blue} 4 \color {red} 98 & \color {red} 7 \color {blue} 47 &  \color {red} 9 \color {red} 96 & \end{array}$

Of the above, $234$, $245$, $246$, $248$ and $249$ have the appropriate $\mathtt {O E}$, $\mathtt {E E}$, $\mathtt {E O}$ and $\mathtt {O O}$ forms, in some order.

We now check the $4$-digit multiples of those $5$ numbers in the same way:


 * $\begin{array} {r|rrrrrrrr} \times & 5 & 6 & 7 & 8 & 9 \\ \hline

234 & \color {red} {1170} & 1404 & 1638 & 1872 & 2106 \\ 245 & 1225 & 1470 & \color {red} {1715} & \color {red} {1960} & \color {blue} {2205} \\ 246 & 1230 & 1476 & \color {red} {1722} & \color {red} {1968} & \color {blue} {2214} \\ 248 & 1240 & 1488 & \color {red} {1736} & \color {red} {1984} & \color {blue} {2232} \\ 249 & 1245 & 1494 & \color {red} {1743} & \color {red} {1992} & \color {blue} {2241} \end{array}$

The numbers of $\mathtt {O O}$ form are marked $\color {red} {\text {red} }$, while those of $\mathtt {E E}$ form are marked $\color {blue} {\text {blue} }$.

It can be seen that of the above, $234$ is eliminated as a candidate as it has no $4$-digit multiple of $\mathtt {E E}$ form.

It remains to explore the individual long divisions which are composed of the above candidate values of $D$ and their multiples.

Since we know the form of each multiple, we can match each form to its corresponding digit of the quotient.

For $D = 246$, the quotient is either $4 \, 071 \, 293$ or $4 \, 081 \, 293$.

However, we would have:
 * $N = D Q \ge 246 \times 4 \, 071 \, 293 = 1 \, 001 \, 538 \,078$

but $N$ only has $9$ digits.

Therefore $D \ne 246$.

The rest of the possibilities all yield solutions, and will be explored in their respective solution pages.