# Even or Odd?

**Number Theory**Level 3

Write a sequence of numbers from 1 to 8192, randomly rearrange these numbers, you'll have \[a_1,~~a_2,~~a_3,~~\ldots,~~a_{8190},~~a_{8191},~~a_{8192}\] Now find the difference of every adjacent two numbers: \[\vert a_1-a_2\vert,~~\vert a_3-a_4 \vert,~~ \ldots,~~\vert a_{8189}-a_{8190}\vert,~~\vert a_{8191}-a_{8192} \vert\] Randomly rearrange the results and you'll have a new sequence of numbers: \[b_1,~~b_2,~~b_3,~~\ldots,~~b_{4094},~~b_{4095},~~b_{4096}\] Find the difference of every adjacent two numbers: \[\vert b_1-b_2\vert,~~\vert b_3-b_4 \vert,~~ \ldots,~~\vert b_{4092}-b_{4093}\vert,~~\vert b_{4094}-b_{4096} \vert\] Again rearrange the results and you'll have another new sequence of numbers: \[c_1,~~c_2,~~c_3,~~\ldots,~~c_{2046},~~c_{2047},~~c_{2048}\] Repeat the process above until you get a number \(x\). then is \(x\) even or odd?

This is one part of 1+1 is not = to 3.