A man places 1 penny on the square of a chess board, 2 pennies on the next square, 4 pennies on the third square, and continues this pattern. How many pennies will there be on the 14th square?

Answer 1

The man places pennies on the squares of a chess board according to a pattern where each square \( n \) contains \( 2^{(n-1)} \) pennies.

1. For square 1, the number of pennies is: \[ 2^{(1-1)} = 2^0 = 1 \text{ penny} \]

2. For square 2, the number of pennies is: \[ 2^{(2-1)} = 2^1 = 2 \text{ pennies} \]

3. For square 3, the number of pennies is: \[ 2^{(3-1)} = 2^2 = 4 \text{ pennies} \]

4. For square 4, the number of pennies is: \[ 2^{(4-1)} = 2^3 = 8 \text{ pennies} \]

Continuing this pattern, we see that for square \( n \), the number of pennies is \( 2^{(n-1)} \).

To find the number of pennies on the 14th square: \[ \text{Number of pennies on square 14} = 2^{(14-1)} = 2^{13} \]

Now, we need to calculate \( 2^{13} \): \[ 2^{13} = 8192 \]

Thus, the number of pennies on the 14th square is: \[ \boxed{8192} \]