SOLUTION: The reverse of a two digit number is the number obtained by writing the digits in reverse order. For example, 37 is the reverse of 73, and 44 is its own reverse. How many three dig

The smallest possible case is 11×11 = 121

Let the the maximal digit be x and the minimal digit be y

x ≧ y

The product is

(10x + y)(10y + x) < 1000
100xy+10x²+10y²+xy < 1000 
   10x²+101xy+10y² < 1000

Let y, the minimal digit, be small as possible, i.e., y=1

   10x²+101x+10 < 1000
   
   10x²+100x-990 < 0

The solution to that quadratic inequality is

-16.14 < x < 6.13

But since x is an integer, x = 1,2,3,4,5, or 6

61×16 = 976, and that is a 3-digit number

So

61×16, 51×15, 41×14, 31×13, 21×12, and 11×11

account for 6 cases.

Let y, be the next smallest digit possible i.e., y=2

   10x²+101xy+10y² < 1000

   10x²+101x(2)+10(2)² < 1000
   
          10x²+202x-40 < 1000

        10x²+202x-1040 < 0     

The solution to that quadratic inequality is

-24.45 < x < 4.25

Since x is an integer, x = 1,2,3, or 4

But 42×24 = 1008, is too large

However,
32×23 = 736 and 22×22 = 484 account for 2 more cases.

That's 8 cases.

The minimal digit y cannot be 3 since 33×33=1089,
which is not a 3 digit number.

So there are only 8 such 3-digit products.

Answer: fewer than 9.  In fact there are only these 8

61×16 = 976
51×15 = 765
41×14 = 574
31×13 = 403
21×12 = 252
11×11 = 121
22×22 = 484
23×32 = 736

Edwin