Overhang

How far off the edge of the table can we reach by stacking <i>n</i> identical blocks of length 1? A classical solution achieves an overhang of 1/2<i>H</i><inf><i>n</i></inf>, where <i>H</i><inf><i>n</i></inf> = Σ^<i>n</i><inf><i>i</i>=1</inf> 1/<i>i</i> ~ ln <i>n</i> is the <i>n</i>^th harmonic number, by stacking all the blocks one on top of another with the <i>i</i>^th block from the top displaced by 1/2<i>i</i> beyond the block below. This solution is widely believed to be optimal. We show that it is exponentially far from optimal by giving explicit constructions with an overhang of Ω(<i>n</i>^1/3). We also prove some upper bounds on the overhang that can be achieved. The stability of a given stack of blocks corresponds to the feasibility of a <i>linear program</i> and so can be efficiently determined.