Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution 1

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Puzzles and Curious Problems by Henry Ernest Dudeney: $110$

An Absolute Skeleton
Here is a good skeleton puzzle.
The only conditions are:
$(1)$ No digit appears twice in any row of figures except the dividend.
$(2)$ If $2$ be added to the last figure in the quotient it equals the last but one,
and if $2$ be added to the third figure from the end it gives the last figure but $3$ in the quotient.
That is to say, the quotient might end in, say, $9742$, or in $3186$.
        ********
    ------------
 ***)***********
      ***
       ---
       ***
       ***
       ----
       ****
       ****
       -----
         ***
         ***
         ----
         ****
         ****
         -----
          ****
          ****
          -----
           ****
           ****
           -----
            ****
            ****
            ----


Solution

        32418675
    ------------
 312)10114626600
      936
       ---
       754
       624
       ----
       1306
       1248
       -----
         582
         312
         ----
         2706
         2496
         -----
          2106
          1872
          -----
           2340
           2184
           -----
            1560
            1560
            ----


Proof

From the initial deductions, we have determined that:

$(1): \quad$ It is possible that there is a solution in which the divisor is $312$
$(2): \quad$ If this is the case, then the quotient is one of:
$32417586$
$32418675$
$32816475$
$32817564$

Thus the dividend in each case will be:

$10114286832$
$10114626600$
$10238740200$
$10239079968$

It remains to investigate each one.


Hence we set up a long division and check the partial dividends in each case. 

        32417586
    ------------
 312)10114286832
      936
       ---
       754
       624
       ----
       1302
       1248
       -----
         548
         312
         ----
         2366
         ****

and we need go no further. 


        32418675
    ------------
 312)10114626600
      936
       ---
       754
       624
       ----
       1306
       1248
       -----
         582
         312
         ----
         2706
         2496
         -----
          2106
          1872
          -----
           2340
           2184
           -----
            1560
            1560
            ----

and we have found a solution.


Let us try the remaining solutions: 

        32816475
    ------------
 312)10238740200
      936
       ---
       878
       ***

and we need not continue with this one.


Finally: 

        32817564
    ------------
 312)10239079968
      936
       ---
       879
       624
       ----
       2550
       ****

and again it is seen that this also does not fit the criteria.


Hence the only solution with $D = 312$ is:

$\dfrac {10114626600} {312} = 32418675$

$\blacksquare$


Sources

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