But the villages, known as chhitmahals, remained marooned. The first failed attempt to resolve the issue came in 1954, when Bangladesh was still part of Pakistan. In 1974, Indira Gandhi and Bangladesh’s prime minister, Sheik Mujibur Rahman, agreed to sort out the border problem, but Mr. Rahman was assassinated before the agreement could be carried out, and the pro-Pakistan government in Bangladesh never followed through. A third attempt in 1992, between India’s prime minister P. V. Narasimha Rao and Khaleda Zia, then the leader of Bangladesh, also went nowhere.

But there are reasons to be more optimistic now. Bangladesh is more stable and prosperous than ever, its economy growing at about 6 percent a year. India’s government, meanwhile, has tried to improve relations with Bangladesh, not least because it has testy relations with almost all of its neighbors, which include territorial disputes.

Muhammad Nazir Hussain, who lives in the enclave of Nalgram, certainly hopes that the question of his citizenship will soon be settled. He lives on land his family has farmed for generations and considers himself Indian. But his village is officially part of Bangladesh. His cousin’s house a few hundred yards away is in India, though half his fields lie in Bangladesh. Even the pond that borders Mr. Hussain’s rice paddy is divided between the two nations, though the ducks that skimmed it did not seem to notice.

“It is a very complicated problem,” he said, with considerable understatement.

Mr. Hussain’s younger brother, Manik Mia, has an Indian voter ID card because he was able to register at the home of a relative in an Indian village. Every family, it seems, is divided in this way.

“If we had been in India, we would have been connected to the road, we would have had a school, health facilities, electricity,” Mr. Mia said. “But we have none of that. At times I wonder, are we human beings or are we animals?”

They are certainly not treated like Indians. In 2006, four men from Madhya Masaldanga were arrested and charged with immigration violations. They had been trying to travel to the northern Indian city of Dehradun to work in the booming construction industry, but they were stopped by the police in a neighboring town. When they could not produce identification they were arrested and eventually convicted and jailed for two years.

Even when they had completed their sentences, the men were not immediately released — the local police told them they were awaiting identity papers for them from Bangladesh. Only after other residents of Madhya Masaldanga mounted a protest were the men released.