A simple mathematics riddle that has me batty.

Question

I'm looking to confirm an answer I came up with. I'm pretty sure this is going to seem really silly to many of you because it's probably very easy for you to understand, but I can't wrap my head around it.

Problem:

You have 23 lights, each with its own switch. Each switch functions like almost all light switches, each switch is either, "ON" or "OFF." How many unique combinations of the lights being "ON" or "OFF" are possible?

Example:

switch #1 is on, all other switches off, is one unique combo. switch #1 and #2 are on, all others off is another unique combo.

My thoughts:

So I'm thinking since each switch has only 2 settings, the way to come up with the total possible combos is $2^{23}$.

So this comes out to a total of $\color{red}{8,388,608}$!

This number seems incredibly high to me. Looking at these 23 switches in front of me (for my kid's school project) I just can't believe there are nearly 8.4 million combos possible.

Any help or confirmatiin is greatly appreciated. Thx in advance. Charles

That's the power of exponential. It grows crazy fast. Adding a new switch doubles the amount of possible settings. So each time you add one it gets doubled, with 24 it's 16'777'216, and you just need 30 to get over a billion. — Kolja, Sep 25 '18 at 21:39
See the wheat and chessboard problem, $8$ million is pocket change ;-) — dxiv, Sep 25 '18 at 22:25
Thx so much for the help. Appreciate you folks taking the time to clarify! — Charles Ar, Sep 25 '18 at 23:46

score 5 · Answer 1 · answered Sep 25 '18 at 21:47

5

Good work.

To get a quick estimate without calculating it's useful to remember that $2^{10}$ is approximately $1000$ since it's exactly $1024$. Then $2^{20}$ is approximately $1000 \times 1000$ which is a million. Then $2^{23}$ is about $2^3 = 8$ times that, or $8$ million.

answered Sep 25 '18 at 21:47

Ethan Bolker

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Thank you for the help – Charles Ar Sep 25 '18 at 23:47

A simple mathematics riddle that has me batty.

1 Answers1