Track / Stage 4 / 244 of 340 #504 of 1964

Problem 504

AMC 12 late, AIME early

Combinatorics Difficulty 4.9 Find the answer HMMT_2

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. Spacing and $ signs are ignored.

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 and solution reproduced as published; topic, difficulty and ordering added by this site.