Block-stacking problem

In statics, the block-stacking problem (sometimes known as The Leaning Tower of Lire (Johnson 1955), also the book-stacking problem, or a number of other similar terms) is a puzzle concerning the stacking of blocks at the edge of a table.

The first nine blocks in the solution to the single-wide block-stacking problem with the overhangs indicated

Statement

The block-stacking problem is the following puzzle:

Place identical rigid rectangular blocks in a stable stack on a table edge in such a way as to maximize the overhang.

Paterson et al. (2007) provide a long list of references on this problem going back to mechanics texts from the middle of the 19th century.

Variants

Single-wide

The single-wide problem involves having only one block at any given level. In the ideal case of perfectly rectangular blocks, the solution to the single-wide problem is that the maximum overhang is given by times the width of a block. This sum is one half of the corresponding partial sum of the harmonic series. Because the harmonic series diverges, the maximal overhang tends to infinity as increases, meaning that it is possible to achieve any arbitrarily large overhang, with sufficient blocks.

NMaximum overhang
expressed as a fractiondecimalrelative size
11/20.5 0.5
 
23/40.75 0.75
 
311/12~0.91667 0.91667
 
425/24~1.04167 1.04167
 
5137/120~1.14167 1.14167
 
649/401.225 1.225
 
7363/280~1.29643 1.29643
 
8761/560~1.35893 1.35893
 
97 129/5 040~1.41448 1.41448
 
107 381/5 040~1.46448 1.46448
 
NMaximum overhang
expressed as a fractiondecimalrelative size
1183 711/55 440~1.50994 1.50994
 
1286 021/55 440~1.55161 1.55161
 
131 145 993/720 720~1.59007 1.59007
 
141 171 733/720 720~1.62578 1.62578
 
151 195 757/720 720~1.65911 1.65911
 
162 436 559/1 441 440~1.69036 1.69036
 
1742 142 223/24 504 480~1.71978 1.71978
 
1814 274 301/8 168 160~1.74755 1.74755
 
19275 295 799/155 195 040~1.77387 1.77387
 
2055 835 135/31 039 008~1.79887 1.79887
 
NMaximum overhang
expressed as a fractiondecimalrelative size
2118 858 053/10 346 336~1.82268 1.82268
 
2219 093 197/10 346 336~1.84541 1.84541
 
23444 316 699/237 965 728~1.86715 1.86715
 
241 347 822 955/713 897 184~1.88798 1.88798
 
2534 052 522 467/17 847 429 600~1.90798 1.90798
 
2634 395 742 267/17 847 429 600~1.92721 1.92721
 
27312 536 252 003/160 626 866 400~1.94573 1.94573
 
28315 404 588 903/160 626 866 400~1.96359 1.96359
 
299 227 046 511 387/4 658 179 125 600~1.98083 1.98083
 
309 304 682 830 147/4 658 179 125 600~1.99749 1.99749
 

The number of blocks required to reach at least block-lengths past the edge of the table is 4, 31, 227, 1674, 12367, 91380, ... (sequence A014537 in the OEIS).[1]

Multi-wide
Comparison of the solutions to the single-wide (top) and multi-wide (bottom) block-stacking problem with three blocks

Multi-wide stacks using counterbalancing can give larger overhangs than a single width stack. Even for three blocks, stacking two counterbalanced blocks on top of another block can give an overhang of 1, while the overhang in the simple ideal case is at most 11/12. As Paterson et al. (2007) showed, asymptotically, the maximum overhang that can be achieved by multi-wide stacks is proportional to the cube root of the number of blocks, in contrast to the single-wide case in which the overhang is proportional to the logarithm of the number of blocks.

Robustness

Hall (2005) discusses this problem, shows that it is robust to nonidealizations such as rounded block corners and finite precision of block placing, and introduces several variants including nonzero friction forces between adjacent blocks.

References
  1. Sloane, N. J. A. (ed.). "Sequence A014537 (Number of books required for n book-lengths of overhang in the harmonic book stapling problem.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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