A dollar sign is formed by drawing two parallel vertical lines through the letter $\mathrm{S}$, as shown. These lines cut the letter $S$ into 7 pieces. What is the minimum total number of parallel vertical lines that are needed to cut the letter S into exactly 154 pieces?
(A) 23
(B) 44
(C) 22
(D) 51
(E) 70

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-061.jpg?height=371&width=144&top_left_y=682&top_left_x=1443)
The first vertical line through the letter $S$ cuts the $S$ into 4 pieces, 2 pieces to the left of the line and 2 to the right.

Each additional vertical line adds 3 new pieces while preserving the 4 original pieces.

The chart below shows the number of pieces increasing by three with each additional line drawn.

| Lines | Pieces |
| :---: | :---: |
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
| 4 | 13 |
| $\vdots$ | $\vdots$ |

We want 154 pieces. Since the first line gives 4 pieces, we require an additional 150 pieces. Since three new pieces are created for each additional line drawn, we need to add $150 \div 3=50$ new lines after the first, or 51 lines in total.

ANSWER: (D)