Henry Ernest Dudeney/Puzzles and Curious Problems/110 - An Absolute Skeleton/Solution 2
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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:
******** ------------ ***)*********** *** --- *** *** ---- **** **** ----- *** *** ---- **** **** ----- **** **** ----- **** **** ----- **** **** ----
Solution
32819764 ------------ 310)10174126840 930 --- 874 620 ---- 2541 2480 ----- 612 310 ---- 3026 2790 ----- 2368 2170 ----- 1984 1860 ----- 1240 1240 ----
Proof
From the initial deductions, we have determined that:
- $(1): \quad$ It is possible that there is a solution in which the divisor is $310$
- $(2): \quad$ If this is the case, then the quotient is one of:
- $32418697$
- $32419786$
- $32816497$
- $32819764$
Thus the dividend in each case will be:
- $10049796070$
- $10050133660$
- $10173114070$
- $10174126840$
It remains to investigate each one.
Hence we set up a long division and check the partial dividends in each case.
32418697 ------------ 310)10049796070 930 --- 749 620 ---- 1297 1240 ----- 579 310 ---- 2696 ****
and we need go no further.
32419786 ------------ 310)10050133660 930 --- 750 620 ---- 1301 1240 ----- 613 310 ---- 3033 **** -----
and we need go no further.
32816497 ------------ 310)10173114070 930 --- 873 620 ---- 2531 2480 ----- 511 *** ----
and we need go no further.
Finally:
32819764 ------------ 310)10174126840 930 --- 874 620 ---- 2541 2480 ----- 612 310 ---- 3026 2790 ----- 2368 2170 ----- 1984 1860 ----- 1240 1240 ----
and we have found a solution.
Hence the only solution with $D = 310$ is:
- $\dfrac {10174126840} {310} = 32819764$
$\blacksquare$