Block-stacking problem

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

In statics, the block-stacking problem (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.

Statement

The block-stacking problem is the following puzzle:

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

History

Paterson et al. 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 $\sum _{i=1}^{N}{\frac {1}{2i}}$ 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 $N$ increases, meaning that it is possible to achieve any arbitrarily large overhang, with sufficient blocks.

N Maximum overhang expressed as a fraction decimal relative size 1 1 /2 0.5 0.5   2 3 /4 0.75 0.75   3 11 /12 ~0.91667 0.91667   4 25 /24 ~1.04167 1.04167   5 137 /120 ~1.14167 1.14167   6 49 /40 1.225 1.225   7 363 /280 ~1.29643 1.29643   8 761 /560 ~1.35893 1.35893   9 7 129 /5 040 ~1.41448 1.41448   10 7 381 /5 040 ~1.46448 1.46448

N Maximum overhang expressed as a fraction decimal relative size 11 83 711 /55 440 ~1.50994 1.50994   12 86 021 /55 440 ~1.55161 1.55161   13 1 145 993 /720 720 ~1.59007 1.59007   14 1 171 733 /720 720 ~1.62578 1.62578   15 1 195 757 /720 720 ~1.65911 1.65911   16 2 436 559 /1 441 440 ~1.69036 1.69036   17 42 142 223 /24 504 480 ~1.71978 1.71978   18 14 274 301 /8 168 160 ~1.74755 1.74755   19 275 295 799 /155 195 040 ~1.77387 1.77387   20 55 835 135 /31 039 008 ~1.79887 1.79887

N Maximum overhang expressed as a fraction decimal relative size 21 18 858 053 /10 346 336 ~1.82268 1.82268   22 19 093 197 /10 346 336 ~1.84541 1.84541   23 444 316 699 /237 965 728 ~1.86715 1.86715   24 1 347 822 955 /713 897 184 ~1.88798 1.88798   25 34 052 522 467 /17 847 429 600 ~1.90798 1.90798   26 34 395 742 267 /17 847 429 600 ~1.92721 1.92721   27 312 536 252 003 /160 626 866 400 ~1.94573 1.94573   28 315 404 588 903 /160 626 866 400 ~1.96359 1.96359   29 9 227 046 511 387 /4 658 179 125 600 ~1.98083 1.98083   30 9 304 682 830 147 /4 658 179 125 600 ~1.99749 1.99749

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

• Hall, J. F. (2005). "Fun with stacking blocks". American Journal of Physics. 73 (12): 1107–1116. doi:10.1119/1.2074007Template:Inconsistent citations {{cite journal}}: Invalid |ref=harv (help)CS1 maint: postscript (link).
• Paterson, Mike; Peres, Yuval; Thorup, Mikkel; Winkler, Peter; Zwick, Uri (2007). "Maximum overhang". arXiv:0707.0093 [math.HO]..

External links

• Weisstein, Eric W. "Book Stacking Problem". MathWorld.