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Problem 288

Combinatorics Difficulty 2.4 Multiple choice CEMC Fermat · Canada · 2025

Twelve lightbulbs are in a row. All lightbulbs are initially
turned off. Angie flips the switch for every 2nd lightbulb. Then Bilal
flips the switch for every 3rd lightbulb. Finally, Chenxhui flips the
switch for every 4th lightbulb. At the end of this process, how many
lightbulbs are turned on?

Pick one

Official solution

In the chart that follows, we number the lightbulbs $1$ to $12$ and represent those that are on with a
$1$ and those that are off with a
$0$ .

$\boldsymbol{1}$
$\boldsymbol{2}$
$\boldsymbol{3}$
$\boldsymbol{4}$
$\boldsymbol{5}$
$\boldsymbol{6}$
$\boldsymbol{7}$
$\boldsymbol{8}$
$\boldsymbol{9}$
$\boldsymbol{10}$
$\boldsymbol{11}$
$\boldsymbol{12}$

At the start (all lightbulbs are off):
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$

After Angie flips every 2nd switch:
$0$
$1$
$0$
$1$
$0$
$1$
$0$
$1$
$0$
$1$
$0$
$1$

After Bilal flips every 3rd switch:
$0$
$1$
$1$
$1$
$0$
$0$
$0$
$1$
$1$
$1$
$0$
$0$

After Chenxhui flips every 4th
switch:
$0$
$1$
$1$
$0$
$0$
$0$
$0$
$0$
$1$
$1$
$0$
$1$

At the end of the process, the lightbulbs numbered $2$ , $3$ , $9$ , $10$ , and $12$ are on, and so $5$ lightbulbs are on.

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