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