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

== by : $110$ ==
;An Absolute Skeleton
== Initial Deductions ==
=== 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 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.
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton