A circular chain consists of 4 gems: a diamond, an emerald, a ruby, and a sapphire. In how many ways can the gems be arranged on the chain?

Answer says that fixing one gem gives you the answer 3*2*1=6. But if our gems are S,R,E and D, not only is SRED = DERS, but also SRED=SDER, for if you look at the chain from the other side (say in a mirror) that would be an equal arrangement. So I think the answer is 3, because in a circular chain of four gems and once one gem is fixed, the answer comes from the different options on the spot opposite to the fixed gem.

Normally on a question like this you simply multiply 4 × 3 × 2 × 1 then you get 24. But the test isn’t so easy and a question you are getting at this skill level certainly won’t be that simple. The trick here is that you can flip over a chain. So a Diamond, Emerald, Ruby, Sapphire chain is the same as a S, R, E, D chain that you can flip over. This means that to solve this question you have to fix one gem and you then only have 3 × 2 × 1 possibilities. The answer is 6