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

by : $110$

 * An Absolute Skeleton

Declarations
It is apparent from the structure of the long division sum presented that no digit of $Q$ may be zero.

We also have, from the additional constraint on $Q$, that:

This also gives us that:

The rule about repeated digits offers similar constraints on $n_2$ to $n_8$ and $p_1$ to $p_8$, and will be used without further comment in the following.

However, note that this does not apply to $n_1$, which in this context is the first $4$ digits of $N$, which has no constraint on the uniqueness of its digits.

We have:

This gives us a firm upper bound on $N$, and we can immediately state:


 * $(1): \quad 10 \, 000 \, 000 \, 000 \le N \le 10 \, 859 \, 999 \, 999$

Hence also:

Thus we have:
 * $902 \le p_1 \le 987$

We have that $p_1$, $p_2$ and $p_4$ each have $3$ digits.

But each of $q_1$, $q_2$ and $q_4$ are distinct.

Hence:

We have that $p_3$, $p_5$, $p_6$, $p_7$ and $p_8$ each have $4$ digits.

But each of $q_3$, $q_5$, $q_6$, $q_7$ and $q_8$ are distinct.

Hence:

That is:
 * $205 \le D \le 329$

Thus we have established bounds on $D$ and $N$.

Hence we can now establish bounds on $Q$:

So we immediately see that $3 \le q_1 \le 5$.

Suppose $q_1 = 5$.

Then:

But we already have that $p_1 \le 987$.

So $q_1 \ne 5$.

Suppose $q_1 = 4$.

Then:

But we already have that $p_1 \le 987$.

So $q_1 \ne 4$.

This directly gives us that $q_1 = 3$.

We also have that:

Hence by the same reasoning:
 * $q_2 < 4$ and $q_4 < 4$

and so either:
 * $q_2 = 1$ and $q_4 = 2$

or:
 * $q_2 = 2$ and $q_4 = 1$

Now let us consider the lower bound and upper bound for $Q$.

Recall:

and that $q_1$ to $q_8$ are unique.

We also have that:

This gives us the limits on $Q$:


 * $31 \, 427 \, 586 \le Q \le 32 \, 918 \, 675$

Now:

Similarly:

Thus we have revised our bounds on $D$:
 * $304 \le D \le 329$

Between $304$ and $329$ are $26$ numbers.

Of these, $4$ have duplicated digits:
 * $311$, $313$, $322$ and $323$

so these cannot equal $D$.

There are sufficiently few of these to compute their multiples to investigate whether they have duplicated digits.

We require that $D k$ have no duplicate digits where $1 \le k \le 3$.

If none of them do, then we require that $D k$ have no duplicate digits for at least $5$ values of $k$ where $4 \le k \le 9$.

By the Pigeonhole Principle, in order to eliminate a candidate for $D$, we need to find just $2$ such multiples with duplicated digits where $4 \le k \le 9$.

We will present them as an array, where the numbers in $\color {red} {\text {red} }$ have duplicated digits.

We bother to go no further with a candidate $D$ once the criteria have been breached.


 * $\begin{array} {r|rrrrrrrr} \times & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline

304 & 608 & 912 & \color {red} {1216} & 1520 & 1824 & \color {red} {2128} & \\ 305 & 610 & 915 & \color {red} {1220} & \color {red} {1525} \\ 306 & 612 & 918 & \color {red} {1224} & 1530 & 1836 & \color {red} {2142} \\ 307 & 614 & 921 & \color {red} {1228} & \color {red} {1353} & \\ 308 & \color {red} {616} & \\ 309 & 618 & 927 & 1236 & \color {red} {1545} & 1854 & 2163 & \color {red} {2472} \\ 310 & 620 & 930 & 1240 & \color {red} {1550} & 1860 & 2170 & 2480 & 2790 \\ 312 & 624 & 936 & 1248 & 1560 & 1872 & 2184 & 2496 & \color {red} {2808} \\ 314 & 628 & 942 & 1256 & 1570 & \color {red} {1884} & 2198 & \color {red} {2512} & \\ 315 & 630 & 945 & 1260 & \color {red} {1575} & 1890 & \color {red} {2205} & \\ 316 & 632 & 948 & 1264 & 1580 & 1896 & \color {red} {2212} & \color {red} {2528} & \\ 317 & 634 & 951 & 1268 & \color {red} {1585} & 1902 & \color {red} {2219} & \\ 318 & \color {red} {636} & \\ 319 & 638 & 957 & 1276 & \color {red} {1595} & \color {red} {1914} & \\ 320 & 640 & 960 & 1280 & \color {red} {1600} & 1920 & \color {red} {2240} & \\ 321 & 642 & 963 & 1284 & 1605 & 1926 & \color {red} {2247} & 2568 & \color {red} {2889} \\ 324 & 648 & 972 & 1296 & 1620 & \color {red} {1944} & \color {red} {2268} & \\ 325 & 650 & 975 & \color {red} {1300} & 1625 & 1950 & \color {red} {2275} & \\ 326 & 652 & 978 & 1304 & 1630 & 1956 & \color {red} {2282} & 2608 & 2934 \\ 327 & 654 & 981 & 1308 & 1635 & 1962 & \color {red} {2289} & \color {red} {2616} & \\ 328 & \color {red} {656} & \\ 329 & 658 & 987 & \color {red} {1316} & 1645 & 1974 & \color {red} {2303} & \end{array}$

As can be seen, there are three candidates for $D$:
 * $310$, $312$, $326$

Suppose $D = 326$.

Then $Q$ is such that it has no $7$.

So $Q$ is such that $q_8, q_8 + 2, q_6, q_6 + 2$ come from $\set {4, 5, 6, 8, 9}$.

It is seen that such a $Q$ is not possible.

Hence $D \ne 326$.

It remains to investigate $310$ and $312$.


 * Let $D = 310$.

Then $Q$ is made from $\set {1, 2, 3, 4, 6, 7, 8, 9}$.

Hence $Q$ can be:


 * $31428697$
 * $31429786$
 * $31826497$
 * $31829764$
 * $32418697$
 * $32419786$
 * $32816497$
 * $32819764$


 * Let $D = 312$.

Then $Q$ is made from $\set {1, 2, 3, 4, 5, 6, 7, 8}$.

Hence $Q$ can be:


 * $31427586$
 * $31428675$
 * $31826475$
 * $31827564$
 * $32417586$
 * $32418675$
 * $32816475$
 * $32817564$

But:
 * $10000000000 = 32051282 \times 312 + 16$

and:
 * $10000000000 = 32258064 \times 310 + 160$

and so it is seen that the candidate values for $Q$ which are smaller than $32000000$ are too small.

Hence we are left with the following to investigate.

For $D = 310$:
 * $32418697$
 * $32419786$
 * $32816497$
 * $32819764$

For $D = 312$:
 * $32417586$
 * $32418675$
 * $32816475$
 * $32817564$

These are to be tested one by one.