Let abcd be nonnegative integers Replace them by ab bc cd d

Let a,b,c,d be non-negative integers. Replace them by

|a-b| , |b-c|, |c-d|, |d-a|

Repeat this process. For exmple

a = 7, b=8, c=2, d=4

7,8,2,4

1,6,2,3

5,4,1,2

1,3,1,3

2,2,2,2

0,0,0,0

Prove that for any choice of a,b,c,d, eventually we will get to 0,0,0,0

Solution

we have a,b,c,d non-negative integers, then |a-b|,|b-c|,|c-d| and |d-a| is also non negative integers. if we take a,b,c,d any choice of numbers |a-b|>=|a|-|b| and |a-b|<=|a|+|b| .Therefore, |a|-|b|<=|a-b|<=|a|+|b| and ||a|-|b||<=|a-b| so Replace this by numbers |b-c| ,|c-d|, |d-a| and adding ||a|-|b||+||b|-|c||+||c|-|d||+||d|-|a||<=|a-b|+ |b-c| +|c-d|+|d-a|, we get to 0