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Sissa's Reward

Age 11 to 14 Challenge Level:
According to an old Indian myth, Sissa ben Dahir was a courtier for a king. Sissa worked very hard and invented a game which was played on a board, similar to chess. The king decided to reward Sissa for his dedication and asked what he would like. Sissa thought carefully and then said, "I would like one grain of rice to be put on the first square of my board, two on the second square, four on the third square, eight on the fourth and so on." The king thought this was a silly request, but little did he know ...

Below is a chess board with the first few squares filled with grains of rice as Sissa asked for:

chessboard with 1 grain of rice in top left square, two grains in adjacent square, four grains in next square

How many grains of rice would there be on the 8th square?

How many grains would you need altogether in order to fill up to the 15th square?

Estimate how many grains you would need in total to fill the entire board in this way. Explain your thinking.
Perhaps Sissa was cleverer than the king thought!