Importance of the Absolute Value

Professor Ducci's observation pertains to the problem of integer iterations on a circle:

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.

Whatever 4 integers you start with, you'll reach the sequence of four zeros after a (very) finite number of iterations. So it may appear surprising that with the removal of the absolute value, the terms of the sequence grow without bound.

Discussion

Let a, b, c, d be the numbers originally placed on the circle. These are replaced with a - b, b - c, c - d, and d - a. Obviously, starting with the first iteration, at any step the sum of the four numbers is zero. We may assume that this is true to start with:

a + b + c + d = 0.

Let α = a - b, β = b - c, γ = c - d, and δ = d - a.

One way of thinking of the magnitude of the four numbers is to consider them as components of a 4-dimensional vector. Indeed, the norm of such a vector grows without bound only if the same happens to at least one of its components; for the norm measures the distance of the vector's end point to the origin. We'll use the Euclidean norm:

	0	= (a + b + c + d)²
		= [a + b + c + d]²
		= a² + b² + c² + d² + 2(ab + bc + cd + da) + 2(ac + bd)
		= (a + c)² + (b + d)² + 2(ab + bc + cd + da)

On the other hand,

α² + β² + γ² + δ² = 2(a² + b² + c² + d²) - 2(ab + bc + cd + da)

Adding the two gives

	α² + β² + γ² + δ²	= 2(a² + b² + c² + d²) + (a + c)² + (b + d)²
		≥ 2(a² + b² + c² + d²),

meaning that the square of the norm at least doubles with each iteration.

Reference

A. Engel, Problem-Solving Strategies, Springer, 1998, p. 3

Absolute Value

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