Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!usc!zaphod.mps.ohio-state.edu!swrinde!elroy.jpl.nasa.gov!decwrl!waikato.ac.nz!canterbury!cosc.canterbury.ac.nz!chisnall From: chisnall@cosc.canterbury.ac.nz (The Technicolour Throw-up) Newsgroups: comp.theory Subject: Re: Partitioning squares into unequal squares Message-ID: <1991May30.094614.926@csc.canterbury.ac.nz> Date: 29 May 91 21:46:12 GMT References: <9105242046.AA15308@athos.cs.ua.edu> Reply-To: chisnall@cosc.canterbury.ac.nz Organization: Computer Science,University of Canterbury,New Zealand Lines: 30 Nntp-Posting-Host: cantua.canterbury.ac.nz From article <9105242046.AA15308@athos.cs.ua.edu>, by borie@ATHOS.CS.UA.EDU (Richard Borie): > Is it possible to partition the interior of a square into > smaller squares, no two of which have the same size? Indeed it is. I found the following in Martin Gardner's "Mathematical Diversions": The best known problem of this type is that of fitting a set of squares no two of which are alike into a larger square without any overlap or leftover space. If we think of the larger square as a lattice of unit squares to be divided along lattice lines into unequal squares, the smallest-known square that can be so divided has a side of 175 units. It can be cut into 24 unequal squares. The reader will find a picture of it on page 206 of "The 2nd Scientific American Book of Mathematical Puzzles & Diversions", in a chapter by William T. Tutte explaining how he and his friends used electrical-network theory to find "squared squares" of this type. In case it sounds like this contradicts Neil Calkin's posting I should reiterate someone else's posting: the 112x112 square is the smallest decomposable square when 'smallest' is measured in terms of the number of pieces required, the 175x175 square is the smallest known when 'smallest' is measured in terms of the length of the sides of the square. -- "Merely corrobarative detail, intended to give artistic verisimilitude to an otherwise bald and unconvincing narrative" -- W.S. Gilbert Name: Michael Chisnall email: chisnall@cosc.canterbury.ac.nz