Pigeonhole Principle

\pi = 3.14159265358979323846264
\text{Let } a_1, a_2,\dots a_{24} \text{ represent the first 24 digits of } \pi
Prove:
(a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24}) \text{ is even}

Proof:

13 Odd digits = {3.14159265358979323846264 }

11 Even digits

\text {12 brackets :}(a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24})

Put 13 odds into 12 brackets, by Pigeonhole Principle, there is certainly one bracket where
(a_j - a_k) \text{ is a difference of 2 odds, which is an even = 2n}

2n multiplies with any number will always give even.
The product of 2n with the other 11 brackets will always be even.

Therefore
(a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24}) \text { is even}

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s