Towers problem

I was asked this question 2 years ago and I forgot how to to solve it.

There are N towers. The towers target each other via lasers, and a tower can choose how many it will target. Prove that at least 2 towers are targeting the same amount of towers.

#Combinatorics #Proofs #Math

Easy Math Editor

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Comments

Hint 1: Pigeon-hole principle(PHP)

Hint 2: There are n pigeons

Hint 3: Can you ever have a tower targeting 0 and tower targeting n-1 at the same time?

Hint 4: There are n-1 holes

Hint 5: PHP your way through it

Matthew Fan - 7 years, 7 months ago

A tower must target at least one other tower hence let the $1$ st tower attack $1$ tower, the $2$ nd $2$ towers and so on. ( If they don't then one $2$ of them attack the same number of towers.) Continuing like this we get that the $n$ th tower attacks $n + 1$ towers which is a contradiction since there are only $n$ towers. Hence $2$ towers must attack the same number of towers. This is application of the Pigeon Hole Principle. You can read about it in the Techniques tab below the page :)

A Former Brilliant Member - 7 years, 7 months ago

Nvm I made a mistake the N th tower attacks only n towers by this logic.

A Former Brilliant Member - 7 years, 7 months ago

Further the contradiction proof still holds though as a tower cannot target itself.

A Former Brilliant Member - 7 years, 7 months ago

ah, thanks for the answers!

Jayce Cesista - 7 years, 7 months ago

What if N=1?

Daniel Leong - 7 years, 7 months ago

A tower can target at most $n-1$ towers because it cannot target itself.However there are $n$ towers so there are at least 2 towers targeting the same amount of towers.

Tan Li Xuan - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$