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Skeletal Division

The question is to find the value(s) of *. All the values may be same or different. for example- one of the * could be 8 while another one could be 4.

P.S. While cleaning my old cupboard, I came across this. This was a puzzle given to me and my friends as a part of some project work some years back. It fascinated us then and now I wanted to share it here. Do try to solve it. It's real fun.

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    $\begingroup$ Is it guaranteed that all leading digits are nonzero (for example, the single-digit result of the first subtraction)? $\endgroup$ – 2012rcampion Jul 31 '16 at 16:37
  • $\begingroup$ @2012rcampion If it's not marked, then you can consider it to be 0 $\endgroup$ – Sid Jul 31 '16 at 16:44
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    $\begingroup$ That doesn't answer my question; can stars represent zero as well? Another example, can the divisor be anything between 0-999, or just 100-999? $\endgroup$ – 2012rcampion Jul 31 '16 at 16:45
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    $\begingroup$ @2012rcampion Yes, stars can represent 0 (But not first stars, for example, 300 is *** not ****) Divisors is 100-999. $\endgroup$ – Sid Jul 31 '16 at 16:46
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The answer is

Divisor: 199. Dividend: 598000572. Solution: 3005028

Explanation:

The first big hint was with line 3 and 4 (calling line 1 the initial equation, and line 9 being 0000)
xxxx−xxx = 5
This means line 3 is between 1000 and 1004, so line 4 is between 995 and 999.
So the divisor is a three-digit number which has a multiple between 995 and 999.
Using this information, and looking at lines 1 and 2, the divisor must be less than 500. So now our current bounds for the divisor are 100 and 500. And since line 4 (constrained between 995 and 999) is a multiple of our divisor (100 to 500), we generate a list of possible options being...
995: 199
996: 498, 332, 249, 166
997: No options
998: 499
999: 333
The difference between the subtraction of line 1 and line 2 is equal to 1, so the possible values of line two are 499 to 598. We then can check to see what options do not have a multiple between this range, removing 498, 332, 249, 333. Which leaves us with 166, 199, 499.
Using lines 7 and 8, the second value of each number, must be a 5. So it must be at least 1500. Since 166×9 < 1500, this rules out 166.
Finally with lines 5 and 6, if you use 499 it is impossible to get a remainder greater than 150, which is required for line 7.
This leaves only 199 left, so using 199, we get a dividend of 598000572 and an answer of 3005028, which satisfies the conditions of the puzzle.

Here's a picture with all the numbers:

Solution

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  • $\begingroup$ Excellent.... Nicely done +1. $\endgroup$ – Sid Aug 1 '16 at 11:30
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    $\begingroup$ Sid, is this a type of puzzle that exists? Or did you and your friend invent it? I am very curious here! $\endgroup$ – Topple Aug 4 '16 at 22:40
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    $\begingroup$ @Topple: I've definitely seen this puzzle before. $\endgroup$ – Deusovi Aug 5 '16 at 3:06
  • $\begingroup$ @Topple Well, we didn't invent this puzzle. It was given to us to solve as part of a project work. I think, you can find some of these on the internet... $\endgroup$ – Sid Sep 15 '16 at 15:59
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Here’s a pretty MathJax rendering of the solution (which I found independently after slimeArmy’s answer was posted). $$ % Define a very small, pale comma: \newcommand{\c}{\hskip{-0.5pt}\large\color{#888},\huge\hskip{-6.5pt}} % If you want to eliminate the commas, just do % \newcommand{\c}{} % % Define a red "5": \newcommand{\5}{\color{#D11}5} % Change the above definition to change the color of all the fives % simultaneously and synchronously. % % Define a small, unobtrusive minus sign, and underline the following number: \newcommand{minus}[1]{^{_{_{_-}}}\underline{#1}} % % This is what the MathJax for the long division looks like: % % \huge\begin{align} % 3\c005\c028 \\ % 199\Huge/\!\huge\overline{\,\598\c000\c572}\\ % \minus{597}\phantom{\c000\c000}\\ % 1000\phantom{\c\c000} \\ % \minus{995}\phantom{\c\c000}\\ % \557\phantom{\c\c0} \\ % \minus{398}\phantom{\c\c0}\\ % 1\592 \\ % \minus{1592}\\ % 0 % \end{align} % % Unfortunately, multi-line MathJax doesn't work in spoiler blocks, % so I have to do this: $$

$$\huge\begin{align}3\c005\c028\\199\Huge/\!\huge\overline{\,\598\c000\c572}\\\minus{597}\phantom{\c000\c000}\\1000\phantom{\c\c000}\\\minus{995}\phantom{\c\c000}\\\557\phantom{\c\c0}\\\minus{398}\phantom{\c\c0}\\1\592\phantom{\c\c}\\\minus{1592}\phantom{\c\c}\\0\phantom{\c\c}\end{align}$$

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