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Super smart kids can do math in their heads and are known for not showing work.
He said that the three of them were very angry and insisting that he admit to cheating, and that they were getting increasingly angry at his denials.
It's low odds that he made multiple, exactly offsetting, mistakes on two problems. And if it were my kid, I'd probably just take the zero and move on.
It also seems, from the original post, likely this is a private school (he has been there 4 years and is now a sophomore; they refer matters to the head of school). Private schools usually don't follow the bureaucratic framework that public schools have to. At our private school, refusing to answer questions would lead to an automatic suspension/expulsion for a student, so no, the above advice to contact one's parents before talking would not work at all. Many private schools have a similar approach-they wish to be able to talk to the student directly, without parents present,and if that is a problem,the student shouldn't enroll.
And these responses explain why teachers don't usually bother dealing with cheating. There is no incentive to do so whatsoever, the parents will deny and raise a ruckus, the administrators will be annoyed-frankly, it's a miracle anyone bothers to raise the cheating issue anymore;no wonder it is so commonplace now. Heaven forbid we hold our kids accountable for their actions.