A chessboard is the type of game board used for the game of chess, on which the pawns and pieces are placed. A chessboard is usually square in shape, with an alternating pattern of squares in two colors, with its side being divided into eight parts, resulting in a total of 64 squares. The inventor of the chessboard suggested a reward of one gram of wheat for the first square, 2 grams for the second, 4 grams for the third, and so on, doubling the number of grains for subsequent squares.

Based on the above information, answer the questions given below:

i) How many grains would have to be given to the inventor for 64 squares?

ii) How many grains would have to be given to the inventor for the 12th square?

The total wheat grains for 64 squares on a chessboard, doubling each time starting from one, is 2^64 - 1 grains. The grains on the 12th square would be 2^11 according to the progression.

Explanation:

This question refers to a well-known mathematical problem, often referred to as the chessboard problem or the wheat and chessboard problem. It's based on an ancient Indian legend, and demonstrates the concept of geometric progression - a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number.

i) To find the total number of grains for 64 squares, we add up all the grains on each square. Since the grains double on each square, this is actually the sum of a geometric series. The formula to calculate the sum S of a geometric series where a is the first term, r is the common ratio, and n is the number of terms is S = a * (1 - r^n) / (1 - r). In this case, a = 1 (one grain on the first square), r = 2 (the grains are doubled each time), and n = 64 (64 squares). Therefore, total grains = 1 * (1 - 2^64) / (1 - 2) = 2^64 - 1.

ii) To find the number of grains on 12th square, we don't need to calculate all the previous squares - we can use the formula for the nth term of a geometric sequence, which is a * r^(n-1) where a is the first term, r is the common ratio, and n is the term number. Here, we're looking for the 12th term, so n = 12. The grains on the 12th square = 1 * 2^(12-1) = 2^11 grains.

Learn more about Geometric Progression here:

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