2009 Grade 8 CEMC Gauss Problems/Problem 22

Problem

A dollar sign is formed by drawing two parallel vertical lines through the letter 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?

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$\text{ (A) }\ 23 \qquad\text{ (B) }\ 44 \qquad\text{ (C) }\ 22 \qquad\text{ (D) }\ 51 \qquad\text{ (E) }\ 70$

Solution

We can see that the first line cuts the letter S into four pieces ( $2$ pieces to the left of the line, and $2$ pieces to the right), and that each additional line adds $3$ more pieces.