Solution

I can't stop patting myself; I have hit the solution. It is 11 x 11 - 1 = 120.

But, Sudhir isn't quite happy; I must find all the solutions, he says.

Well, we have ** x ** - * = 120.

Thus, ** x ** = 120 + *.

Now, each star is a digit ranging from 1 to 9. Therefore, ** x ** = 120 + * can be any of these:

** x ** = 120 + 0 = 120

** x ** = 120 + 1 = 121

** x ** = 120 + 2 = 122

** x ** = 120 + 3 = 123

** x ** = 120 + 4 = 124

** x ** = 120 + 5 = 125

** x ** = 120 + 6 = 126

** x ** = 120 + 7 = 127

** x ** = 120 + 8 = 128

** x ** = 120 + 9 = 129

Let's consider each of these:

Now, 120 = 2 x 60 = 2 x 2 x 30 = 2 x 2 x 2 x 15 = 2 x 2 x 2 x 3 x 5. From this, I do get two two-digit numbers: 12 and 10. Thus, 12 x 10 - 0 = 120 is one solution.

Now, 121 = 11 x 11. Thus, 11 x 11 - 1 = 120 is another solution.

Next, consider 122. Now, 122= 2 x 61. I don't get two two-digit numbers.

It's the turn of 123. Well, 123 = 3 x 41. I don't get two two-digit numbers.

Now to tackle 124. In this case, 124 = 2 x 2 x 31. I don't get two two-digit numbers.

Come on, 125. Now, 125 = 5 x 5 x 5. No two two-digit numbers here too.

What about 126? This looks promising because 126 = 2 x 3 x 3 x 7. But, no! All possible combinations fail to give me two two-digit numbers here too.

Next is 127. In case of 127, there are no divisors other than 1 and 127 itself. So, there is no question of finding two two-digit numbers here.

Let's get on to 128. This too looks promising because 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2. Great! There are seven 2s; that should give me two two-digit numbers. No! The first two-digit number I hit upon is 16, but that leaves me with 8 - a single-digit number. Failure, here too!

At last, I reach 129. But, 129 = 3 x 43 and there are no two two-digit numbers.

Thus, there are two solutions: 11 x 11 - 1 = 120, and 12 x 10 - 0 = 120.