By Daniel J. Velleman

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**Extra info for American Mathematical Monthly, volume 116, number 1, january 2009**

**Example text**

2. 4. D(n) ≥ ( 3n )1/3 − 16 1 4 for all n. Proof. Choose d so that (d − 1)d(2d − 1) d(d + 1)(2d + 1) +1≤n ≤ . 3 shows that a d-stack yields an overhang of d/2 and can be constructed using n or fewer blocks. Any extra blocks can be just placed in a vertical pile in the center on top of the stack without disturbing balance (or arbitrarily scattered on the table). Hence 2(d + 12 )3 n< 3 and so D(n) ≥ d/2 > 3n 16 1/3 1 − . 4 In Section 3 we claimed that optimal stacks are spinal only for n ≤ 19. 7 deals with the range n ≥ 5000.

San Francisco, 1971. 8. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley Longman, Reading, MA, 1988. 9. J. F. Hall, Fun with stacking blocks, Amer. J. Phys. 73 (2005) 1107–1116. 10. J. E. Hearnshaw and M. S. Paterson, Problems drive, Eureka 27 (1964) 6–8 and 39–40. html. 11. C. P. Jargodzki and F. Potter, Mad About Physics: Braintwisters, Paradoxes, and Curiosities, John Wiley, New York, 2001. 12. P. B. Johnson, Leaning tower of lire, Amer. J. Phys. 23 (1955) 240.

A schematic description of a well-behaved set of balancing forces. A useful property of well-behaved collections of balancing forces is that the total weight of the stack and the positions of its blocks uniquely determine all the forces in the collection. This follows from the fact that each block has either two downward forces acting upon it at specified positions, namely at its two edges, or just a single force in an unspecified position. Given the upward forces acting on a block, the downward force or forces acting upon it can be obtained by solving the force and moment January 2009] OVERHANG 39 equations of the block.