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.
You... you are telling me... that I could break that damned chain more, than 3 bucking times??!! YOU JUST SHOWED A SOLUTION WITH SEVEN BREAKS PER CHAIN??!! MY LOVELY FIVE BREAKES COULD HAVE WORKED JUST FINE!!! BY BEING UNCLEAR WITH THE CONDITIONS YOU SPENT SO MUCH OF MY F-F-F... IREWORKING TIME!!! DO YOU EVEN REALISE, HOW MUCH YOU COULD HAVE SPARED ME x9999?!!
9356661
Um...no. I was explaining where the three breaks in the chain should be. Read it again.
As said in the chapter:
Three breaks. Which happen between the links. Now let's count (coloured for convenience): 3 + 1 + 1 + 1 + 1 = 7, and this so happens to be just three more, than 4 -- that's, how many parts we get with just 3 breaks.
Your answer into consideration, by 3 breaks you meant "take out no more, than 3 single links"... Did you read my answer? I mentioned, that, had Fluttershy been allowed to make more breaks, she could have broken the chain into six parts, thus making 5 breaks -- but not taking out 5 single links! In fact, no links would have been taken out! I had the solution from the very beginning, and only due to unclear definition of what "break" was supposed to be did I spend, if I may repeat, so much damned time! Had you clearly stated, that "break" meant "single link taken out"...
9357076
It doesn't take out any links. The broken three links are your first three payments. On the fourth, you take them back and hand over the four link chain. Then the three broken links are handed back for the fifth, sixth and seventh payment respectively, and so on.
Your solution also involves having to pay in advance, or after the job.
You have to go with my solution, which allows her to pay one link at a time.