Question

Imagine letters are extruded into three-dimensional solids (like the block letter A shown). If the word LOVE is extruded in the same way, how many surfaces would the resulting object have? 

Hint:

For any extruded flat shape: \(\textSurfaces = 2 +\) (number of boundary runs across all outlines). Count straight-run changes at corners; count one continuous side for each curved run; add two for front and back. Sum letter-wise for a whole word.

Answer 1

Extruding a flat letter shape of uniform thickness produces a solid with: one front face and one back face (two planar faces), and one separate side face for each straight edge run around the 2D outline (outer boundary and, if present, inner holes). Curved runs (like the circular part of O) contribute a single continuous curved side face per run.
So, for each letter: \[ \text{surfaces} \;=\; 2 \;+\; (\text{number of boundary runs around all outlines}). \] We now walk the outlines of the four block letters. The counts below follow one full clockwise trace of each letter’s boundary; every time the direction changes at a corner, a new planar side face begins.
(L) — a rectilinear “step” shape with eight straight boundary runs (down, right, up, left, up, left, down, right).
Side faces $=8$, plus front and back $=2 ⇒$ \(\boxed{10}\) faces.
(O) — an annulus (one outer smooth curve + one inner smooth curve).
Side faces $=2$ (outer cylindrical surface $+$ inner cylindrical surface), plus front and back $=2 ⇒$ \(\boxed{4}\) faces.
(V) — a block “chevron”. Its polygonal outline has seven straight runs (two long outer slants, two short top returns, two inner slants forming the notch, and the base).
Side faces $=7$, plus front and back $=2 ⇒$ \(\boxed{9}\) faces.
(E) — a rectilinear outline with the vertical stem and three horizontal arms; the perimeter breaks into twelve straight runs (each arm contributes a pair of small verticals and a tip segment in addition to the stem’s runs).
Side faces $=12$, plus front and back $=2 ⇒$ \(\boxed{14}\) faces.
Total surfaces for LOVE
\[ 10\;(\text{L}) + 4\;(\text{O}) + 9\;(\text{V}) + 14\;(\text{E}) \;=\; \boxed{35}. \]