The only true synonyms in English?

I was playing hide and seek with the kids in the local park yesterday and I got talking to one of the other parents. Hearing that I worked with English, he asked me if I knew  which words were the only completely interchangeable synonyms in English. It was something he’d learnt in school and never forgotten.

Of course I didn’t.

The answer was gorse, furze and whin. They are interchangeable words for ulex europaeus: a common shrub.

Are there really no other synonyms of this kind in English?

I was a littlesuspicious so I googled just to be sure. On a blog, one wag had come up with these suggestions:

Nil and zero, twelve and dozen and  selfish and Republican!