You know wherever the three breaks in the chain are, you have three single broken links. Those will at least take care of your first three payments.
Continuing the thought experiment, you realise that you'll need a length of chain four links long to make a fourth payment, since you're only allowed three breaks. So snap the fifth link in the chain.
With that four-link chain and the three single links, those cover up to the seventh payment. Now, when the time comes to make an eighth payment, you need an eight-link chain, taking back the first seven links. So you snap the ninth link of your now fifty-eight link chain.
You've now used up fifteen links so far, so when the time comes to make a sixteenth payment...you guessed it. Snap the seventeenth link of your now forty-nine link chain.
That now leaves you with three single broken links, a four-link chain, an eight-link one, a sixteen-link one and a thirty-two link one. From there, you just keep swapping the links backwards and forwards as needed to keep the payments up.