3. An old story tells us that the inventor of chess was told by his ruler that h
ID: 3114180 • Letter: 3
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3. An old story tells us that the inventor of chess was told by his ruler that he could have any reward he wished. He asked for a grain of wheat on the first square of the chess board, two grains on the second, twice that on the third square, and then double the number of grains for each successive square. It turned out that his reward would have bankrupted the kingdom. Consider that legend, and think about the question: would you rather have a million dollars or a chess board with a penny on the first square, two cents on the second square, twice that on the third square, and then double the number of pennies on each successive square? (There are 64 squares on a chess board.) a. How much would the money on the 1oth square be worth? b. Which square would be the first one to have pennies worth more than a million dollars?Explanation / Answer
A penny is worth one cent. The description entails a geometric series with 1 as the first term and 2 as the common ratio. The nth term of this geometric series is 1*2n-1.
a. The money on the 10th square will be worth 210-1 pennies/cents or 29 = 512 cents = $ 5.12.
b. If the nth square is the first one to have pennies worth more than 1 million dollars, then we have 1/100(2n-1) > 1000000 or, 2n-1 > 100000000 or, 2n-1 > 108 . Now, on taking logarithm of both the sides, we get (n-1)log2 > 8 log 10 ( as log ab = b log a ) or, n-1 > 8/log 2 or, n > 1+8/(0.301029995) or, n > 1 + 26.57542482 or, n > 27.57542482. Thus, the 28th square is the first square which will have pennies worth more than $ 1 million.
Now, the answer of the first question is apparent. A chessboard has 64 squares and the 28th square itself will have pennies worth more than a million dollars. Hence, the option with a penny on the first square , two cents on the 2nd square, twice that on the 3rd square, and then double the number of pennies on each successive square will entail a sum much higher than 1 million dollars, and therefore, this option will be preferred.
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