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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:
\[
\boxed{8192}
\]
make this like a 7th grader wrote it

1 answer

Okay, let's break it down in a way that a 7th grader might explain it!

So, there is this cool pattern for placing pennies on a chessboard. Each square on the board gets a certain number of pennies depending on its number. Here's how it works:

  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: \[ 2^{(4-1)} = 2^3 = 8 \text{ pennies} \]

And it keeps going like that! Basically, for each square number \( n \), you have \( 2^{(n-1)} \) pennies.

Now, let’s figure out how many pennies are on square 14. We do this by plugging 14 into our formula:

\[ \text{Number of pennies on square 14} = 2^{(14-1)} = 2^{13} \]

Now we need to calculate \( 2^{13} \).

Here's how \( 2^{13} \) works:

  • It means you multiply 2 by itself 13 times, or: \[ 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \]

And when you do that, you get: \[ 2^{13} = 8192 \]

So, on the 14th square, there are 8192 pennies!

So the answer is: \[ \boxed{8192} \]