Potty trained by one...whose trained the child or the parent? Some children are not walking at one - you wait for the signs and take the child to the restroom. It may work for some but the reality is that the child is the only one that can do this. They need to be able to recognize and understand their body telling them they need to go and have the ability to control when they have to urinate or have a bowel movement (to "Hold it"). There is no link between a child's age, intelligence and ability to be potty trained. In fact, if you look at it this is the first thing a child has absolute control over - we pick what they eat, what they wear, when they go to bed etc. - they in turn choose when they are ready.
Some of the things that worked for us with my daughter (she was 3yrs 3mos as she was ready and able to stay dry day and night) We used M&M's so many for urinating and another amount for bowel movements - both at the same time bonus add those numbers together (when out and about just kept track actually she kept track very closely) One friend who wanted to use the same reward approach used tickets as her son's currency was hot wheels, the tickets were collected and traded for the litte cars. When we had an "accident" we just encouraged her try harder next time and made a huge deal on her successes. I have friends that have had great success with Dr. Phil's program too.
I have another to potty train - my son (just turned 2) and he seems to have a huge interest in going potty - he will ask to sit on the potty and goes when he is on there. Both of my children prefer to use the "big potty" with the seat insert perhaps yours may too (he may want to be able to pick it out) New undies that were picked out by my daughter and left in the package until she was done training was a big interest for her too.
I wish you luck and patience. It is very hard not to compare our children but as with every past, present and future doctor, president, rocket scientist, peace leader, mother etc. they, too, were potty trained when they were ready...so will yours. :o)