Yes, there is a known regular way to tile a (1 \times 1) square with non-repeated rectangles of the form (\frac{1}{n} \times \frac{1}{n+1}) for (n = 1, 2, 3, \ldots). This tiling leverages the telescoping nature of the series (\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1) and recursively arranges each rectangle to manage the remaining area.

Key Insight:

The series (\sum_{n=1}^{\infty} \f...

Yes. There is a simple “corner-greedy” tiling that uses each rectangle 1/n by 1/(n+1) exactly once and fills the unit square.

Construction (axis-parallel, using each n once, up to rotation):

• Step 1: Put R1 = 1 × 1/2 as a strip along the bottom of the unit square.
• Inductive step: Suppose you have placed R1, R2, …, Rn so that the unfilled region is an orthogonal L-shaped region with a single in...

Up to the present day nobody knows the answer – the problem is open.

The identity

  Σ_{n≥1} 1/[n(n+1)] = 1

only tells us that the total area of the prescribed rectangles equals the area of the unit square; it does not give any information about the geometry that is needed for a tiling. People have tried to exploit the telescoping partial‐fraction 1/[n(n+1)] = 1/n – 1/(n+1), the fact that every...

No systematic (i.e. “regular’’) tiling of the unit square that uses every rectangle

 R n = (1 / n) × (1 / ( n + 1)) , n = 1 , 2 , 3 , … , once and only once is known.
Neither is there a theorem that such a tiling is impossible.
The problem is still open.

What is known.

1. In every such rectangle the longer side is ≥ ½ and the shorter side is ≤ ½, and for n > 2 both sides are < ½.
   ...

Excellent question! This is a classic problem in recreational mathematics that beautifully connects an infinite series to a geometric tiling puzzle. The sum of the areas is indeed 1, which is a necessary condition, but the possibility of actually fitting the pieces together without overlap is a much deeper question.

The answer is yes, there are several known "regular ways" to tile the square....

Regular Tiling of a Unit Square with 1/n × 1/(n+1) Rectangles

The question asks for a systematic way to tile a 1×1 square with non-repeated rectangles of dimensions 1/n × 1/(n+1), motivated by the fact that:

A Systematic Approach

Yes, there is a known regular way to tile the unit square with these rectangles! The most elegant approach uses a rec...

This is a fascinating question about tiling a unit square with rectangles of dimensions .

The mathematical foundation is solid: since , the total area of all these rectangles equals the area of the unit square. However, the geometric challenge of actually arranging them to perfectly tile the square is non-trivial.

**Key O...

First, I should understand the dimensions. Each rectangle is (...

Yes – in fact one can carry out a perfectly mechanical “greedy‐Euclid” procedure which at the th step takes your empty rectangular remainder and peels off exactly the ‐tile from its largest side. One checks inductively that at step the remaining region is again a rectangle, of area and that its longer side is at least $...