Suppose the first sign had the two-digit number "AB".
   The value is actually:  10A + B

An hour later, the second sign had the number "BA".
   The value is actually:  10B + A

The difference is the distance he drove in an hour.
   (10B + A) - (10A + B)  =  9B - 9A

This is also his speed:  (9B - 9A) miles per hour.


Another hour passes and the sign says "A0B" or "B0A".
   The values are:  100A + B  or  100B + A, respectively.

The difference of the third number and the second number is also his speed.


Suppose the third sign said "B0A" or 100B + A.
Then his speed is:  (100B + A) - (10B + A) =  90B

We've expressed her speed in two ways; they must be equal.
   9B - 9A  =  90B   ÂÂ¨   -9A  =  81B   ÂÂ¨   A  =  -9B

But both A and B are positive digits.
So this case is impossible.


The third sign must have said "A0B":  100A + B
   The difference is:  (100A + B) - (10B + A)  =  99A - 9B

This equals his speed:  9B - 9A  =  99A - 9B
   then:  18B =  108A   ÂÂ¨   B = 6A


Since A and B are digits:  A = 1, B = 6

The first sign had "16", the second had "61", the third had "106"

his speed is 45 miles per hour.