By the time they got the last offer, Quinn and Daryn Shapurji had received 54 bids on their four-bedroom, single-family house in Fishers, Ind., in just three days. Ms. Shapurji said they felt totally overwhelmed — and a bit melancholy.

“We felt bad that we had to say no to so many people, because we got a lot of beautiful letters from buyers saying how much they loved our house and why they wanted to live in the area,” said Ms. Shapurji, 32, a closing coordinator for a home builder. “Some buyers had already struck out on five or six homes.”

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