Question

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

Answer 1

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:
\[
\boxed{8192}
\]