A New Development of the Classical Single Ladder Problem via Converting the Ladder to a Staircase

Ralph Høibakk,Annette Meidell,Dag Lukkassen,Lars Erik Persson

doi:10.3390/math9040339

Ralph Høibakk, Annette Meidell + Show 2 more

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https://doi.org/10.3390/math9040339

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Abstract

Our purpose is to shed some new light on problems arising from a study of the classical Single Ladder Problem (SLP). The basic idea is to convert the SLP to a corresponding Single Staircase Problem. The main result (Theorem 1) shows that this idea works fine and new results can be obtained by just calculating rational solutions of an algebraic equation. Some examples of such concrete calculations are given and examples of new results are also given. In particular, we derive a number of integer SLPs with congruent ladders, where a set of rectangular boxes with integer sides constitutes a staircase along a common ladder. Finally, the case with a regular staircase along a given ladder is investigated and illustrated with concrete examples.

Highlights

It is well known that the solutions to the quartic equations2021, 9, 339. https://doi.org/10.3390/math9040339Academic Editors: Alberto Cabada and William GuoReceived: 29 April 2020Accepted: 9 December 2020Published: 8 February 2021Publisher’s Note: MDPI stays neutral z4 − 2Lz3 + ( L2 − a2 − b2 )z2 + 2La2 z − L2 a2 = 0 (1)give solutions to the Single Ladder Problem (SLP)
The Single Ladder Problem was a very popular subject in both professional and recreational mathematics and a number of papers have been published demonstrating real solutions to different versions of the problem; see, e.g., [7,8,9,10,11,12,13,14,15] and the recent especially illuminating review article [16] by Coxeter. Several of these developments have concentrated on the attempt to find integer solutions of the SLP
The purpose of this paper is to continue the Diophantine avenue initiated in [17] and to investigate: (Section 2): If a number of integer rectangular boxes with sides ai and bi can touch a common integer ladder L with a fixed position in quadrant one, constituting a staircase along the ladder, see Figure 4; (b) (Section 3): If the variables ai and bi of the rectangular boxes defined in (a) can be adjusted to constitute a ladder with regular intervals along the common ladder, see Figure 5; and (c) (Section 4): If a number of integer rectangular boxes with variables ai and bi can individually touch integer ladders Li with identical lengths but in different geometrical position in quadrant one; see Figure 6. In a sense this means that the Single Ladder Problem is converted into a corresponding staircase problem we have chosen to call the “Single Staircase Problem”

Summary

Introduction

The Single Ladder Problem was a very popular subject in both professional and recreational mathematics and a number of papers have been published demonstrating real solutions to different versions of the problem; see, e.g., [7,8,9,10,11,12,13,14,15] and the recent especially illuminating review article [16] by Coxeter Several of these developments have concentrated on the attempt to find integer solutions of the SLP. In a sense this means that the Single Ladder Problem is converted into a corresponding staircase problem we have chosen to call the “Single Staircase Problem”

Integer Solutions of SLPs with Rectangular Boxes Touching a Given SLP Ladder

The Main Result for This Case

Infinite Regular Staircase along a Given Ladder

Different SLPs with Ladders of Equal Length

Conclusions