Shrinking Squares. Emipirical exploration

Question

In the Arthur Engel's book "Problem Solving Strategies" there is a particular problem which he calls Shrinking Squares , empirical exploration. The Problem is quite challenging but after reading the proof, it seems pretty straightforward, but then the problem gets even harder and suggests that the property of the presented algorithm stands for nonnegative real numbers too, which perplexed me and didn't find a way to figure how could this be true. The problem starts like this : We have a square, each of its vertices are labeled with a value, forming a quadruple $(a,b,c,d)$. We generate a sequence which goes as follows:$$S_0=(a,b,c,d)$$$$S_{n+1}=T(S_n)=(\lvert a_n-b_n\rvert,\lvert b_n-c_n \rvert,\lvert c_n-d_n \rvert,\lvert d_n-a_n \rvert)$$ The following algorithm will eventually stop at $(0,0,0,0)$ since after at most $4k$ steps all components have to be divisible by $2^k$, and we have for $\forall i \in \mathbb N$, $max(S_{i+1})\le max(S_i)$. Beyond the natual number, Arthur Engel suggests that this following property stands for the real nonnegative numbers gives the following example : $$\sqrt{2}\qquad\qquad\qquad\pi\qquad\qquad\qquad\sqrt{3}\qquad\qquad\qquad e$$$$\pi-\sqrt{2}\qquad\quad\quad\pi-\sqrt{3}\qquad\quad\quad e-\sqrt{3}\qquad\quad\quad e-\sqrt{2}$$$$\sqrt{3}-\sqrt{2}\qquad\quad\quad\pi-e\qquad\quad\quad\sqrt{3}-\sqrt{2}\qquad\quad\quad\pi-e$$$$\pi-e-\sqrt{3}+\sqrt{2}\quad\pi-e-\sqrt{3}+\sqrt{2}\quad\pi-e-\sqrt{3}+\sqrt{2}\quad\pi-e-\sqrt{3}+\sqrt{2}$$$$0\qquad\qquad\qquad0\qquad\qquad\qquad0\qquad\qquad\qquad0$$ The question is how to prove the termination of this algorithm for real positive nonnegative numbers without going through all of the order cases (which are simplified to $2^3$ since the life expectancy of the quadruples is invariant under rotation)?

Answer 1

The claim "suggests that the property of the presented algorithm stands for nonnegative real numbers too" is false.

Given that, find an example.

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The crux is to view the rule $ S(a, b, c, d) = ( |a-d| , |a-b| , |b-c | , |c-d| )$ as a linear transformation not on $ \mathbb{R}^4$, but on

the set of vectors in $\mathbb{R}^4 $ that have strictly decreasing components, for which $S_T (a, b, c, d) = ( a -d , a - b, b - c , c - d ) $.

(Yes, after checking suitable details like what happens when 2 values are equal, for which it ends after at most 6 steps, so we may consider strictly decreasing components.)

You can then apply standard linear algebra to show that there is a positive real eigenvalue, and the corresponding eigenvector is our infinite game.

A good treatment is in Paul Sally's "A Vertical Development of Mathematical problems", Chapter 1.