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Problem 1176

AMC 12 late, AIME early

Combinatorics Difficulty 4.9 Find the answer HMMT February

For the following planar graphs, determine the number of vertices $v$ , edges $e$ , and faces $f$ : i. $v=9, e=16, f=9$ ; ii. $v=7, e=14, f=9$ . Verify Euler's formula $v-e+f=2$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

i. For the graph with $v=9, e=16, f=9$ , Euler's formula $v-e+f=2$ holds as $9-16+9=2$ . ii. For the graph with $v=7, e=14, f=9$ , Euler's formula $v-e+f=2$ holds as $7-14+9=2$ .

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.