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

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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

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Single-wide

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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.

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
 

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

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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. However, it has been shown that in reality this is impossible and the number of blocks that we can move to the right, due to block stress, is not more than a specified number. For example, for a special brick with h = 0.20 m, Young's modulus E = 3000 MPa and density ρ = 1.8×103 kg/m3 and limiting compressive stress 3 MPa, the approximate value of N will be 853 and the maximum tower height becomes 170 m.[2]

Proof of solution of single-wide variant

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The above formula for the maximum overhang of   blocks, each with length   and mass  , stacked one at a level, can be proven by induction by considering the torques on the blocks about the edge of the table they overhang. The blocks can be modelled as point-masses located at the center of each block, assuming uniform mass-density. In the base case ( ), the center of mass of the block lies above the table's edge, meaning an overhang of  . For   blocks, the center of mass of the  -block system must lie above the table's edge, and the center of mass of the   top blocks must lie above the edge of the first for static equilibrium.[3] If the  th block overhangs the  th by   and the overhang of the first is  ,[4]

 

where   denotes the gravitational field. If the   top blocks overhang their center of mass by  , then, by assuming the inductive hypothesis, the maximum overhang off the table is

 

For   blocks,   denotes how much the   top blocks overhang their center of mass  , and  . Then, the maximum overhang would be:

 

 
Mike Paterson's method to increase the overhang of 16 blocks of unit width and breadth b to 21 + b ² by offsetting the blocks perpendicular to their lengths in a diamond formation [5]

Robustness

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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

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  1. ^ Sloane, N. J. A. (ed.). "Sequence A014537 (Number of books required for n book-lengths of overhang in the harmonic book stacking problem.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ Khoshbin-e-Khoshnazar, M. R. (2007). "Simplifying modelling can mislead students". Physics Education. 42: 14–15. doi:10.1088/0031-9120/42/1/F05. S2CID 250745206.
  3. ^ Cazelais, Gilles. "Block stacking problem" (PDF). Archived from the original (PDF) on December 4, 2023.
  4. ^ Joanna (2022-04-14). "The Infinite Block Stacking Problem or the Leaning Tower of Lire". Maths Careers. Retrieved 2023-12-04.
  5. ^ M Paterson et al., Maximum Overhang, The Mathematical Association of America, November 2009
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