Question

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:
\[
\boxed{8192}
\]