One of the problems faced by India soon after Independence was the issue of persuading the Princely States to join the Indian Union.

Explanation:
1. Wealth and debts to be equally distributed among India, Pakistan, and Great Britain: This was not a problem faced by India soon after Independence. Instead, the issue of wealth and debts was a concern during the partition of assets and liabilities between India and Pakistan. The division of financial resources and debts was a complex task, but it was not directly related to the problem faced by India in persuading the Princely States to join the Indian Union.

2. Persuading the Princely States to join the Indian Union: This was a significant challenge for India after Independence. The Princely States were autonomous entities within the British Indian Empire, and they were given the option to join either India or Pakistan or remain independent. The integration of these states into the Indian Union was crucial for the consolidation of the country. The process involved negotiations, discussions, and sometimes even military action to ensure the accession of these states.

3. Economic development: Economic development was indeed a challenge faced by India after Independence. The country had a vast population, widespread poverty, and limited industrial infrastructure. The government had to focus on various aspects of economic development, such as agriculture, industry, infrastructure, and poverty alleviation. Initiatives like Five Year Plans were implemented to promote growth and development in different sectors of the economy.

4. Refugees from Pakistan: The partition of India and Pakistan resulted in massive population movements and the displacement of millions of people. Many Hindus and Sikhs migrated from Pakistan to India, while Muslims migrated from India to Pakistan. The influx of refugees posed a humanitarian crisis and a challenge for the Indian government. It required the provision of basic necessities, resettlement, and rehabilitation of the displaced population.

In conclusion, the problem of wealth and debts being equally distributed among India, Pakistan, and Great Britain was not a challenge faced by India soon after Independence. Instead, the major issues were persuading the Princely States to join the Indian Union, economic development, and dealing with the influx of refugees from Pakistan.