I don't think you should have to cajole a very intelligent 4 y.o. to use the toilet. If he's having consistent accidents then using the toilet periodically shouldn't be a choice - he should be required to go every two hours.
The math challenges relate to number 1's and is a way to get him to go willingly, by himself. Taking his word for it, that he didn't need to go, resulted in wet pants. an example would be: "X times 7 minus 9 equals 40. What's X." The focus on the maths takes his mind off the oppositional frame of mind and he goes happily. We put a times table in there which made it a place to 'do maths'. Like all such tricks however, this one is losing power.
Having said that, he's improved since my last post in respect to weeing but if asked directly will almost always claim he doesn't need to go.
Pre-school was going great for a while but some changes there have shifted the dynamic and he's showing signs of anxiety again, (running about constantly, ignoring people), and sadly has bitten a kid. It's hard to accept just how long term and complex his anxieties and reactions can be, so there's no easier answer as to why things have gone South.
He's been accepted into a selective school for next year and the principle says he hasn't seen a kid like him in 20 years but says he's ready for the challenge.
