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Problem 72

Geometry Difficulty 1.2 Multiple choice CEMC Pascal · Canada · 2026

Five paths from $A$ to $B$ are shown. In which figure is the path
the shortest?

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Pick one

Official solution

In each of the figures 1, 2, 3, and 5, the path between $A$ and $B$ travels exactly $4$ squares horizontally (left to right)
and $4$ squares vertically (bottom to
top), and so each path has length $4+4=8$ . In Figure 4, the path travels
$2$ squares horizontally, $2$ squares vertically, and along the
diagonal of a $2$ by $2$ square. The path along the diagonal of
a $2$ by $2$ square is shorter than travelling along
both a horizontal side and a vertical side of the $2$ by $2$ square. Thus the length of the path in
Figure 4 is less than $8$ and is the
shortest of the given paths. (Using the Pythagorean theorem, we can
confirm that $\sqrt{2^2+2^2}$ is
indeed less than $4$ .)

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.