# What are the missing numbers?

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.

• Is it guaranteed that all leading digits are nonzero (for example, the single-digit result of the first subtraction)? – 2012rcampion Jul 31 '16 at 16:37
• @2012rcampion If it's not marked, then you can consider it to be 0 – Sid Jul 31 '16 at 16:44
• 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? – 2012rcampion Jul 31 '16 at 16:45
• @2012rcampion Yes, stars can represent 0 (But not first stars, for example, 300 is *** not ****) Divisors is 100-999. – Sid Jul 31 '16 at 16:46

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:

• Excellent.... Nicely done +1. – Sid Aug 1 '16 at 11:30
• Sid, is this a type of puzzle that exists? Or did you and your friend invent it? I am very curious here! – Topple Aug 4 '16 at 22:40
• @Topple: I've definitely seen this puzzle before. – Deusovi Aug 5 '16 at 3:06
• @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... – Sid Sep 15 '16 at 15:59


\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}