Integer Iterations on A Circle II

Place 4 integers on a circle. Proceed in iterations. At every step, compute all differences of pairs of consecutive numbers. Place the absolute values of these differences between the corresponding numbers and remove the numbers themselves. Show that after performing the step several times all the numbers will become 0.

For what numbers - other than 4 - the same holds true?

Please try switching to IE 11 (Windows) or Safari (Mac), for no other browser nowadays runs Java applets. If asked whether to allow the applet to load, click Yes - the applet is signed with a security certificate from a trusted company. But, regardless, there is an explanation below.
Integer Iterations on A Circle II

What if applet does not run?

Numbers on the circle can be set either randomly or by modifying each of the circled entries. To modify a number, click a little to the right or left of its centerline.

Explanation

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