Daniel J. Velleman's American Mathematical Monthly, volume 116, number 1, january PDF

By Daniel J. Velleman

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5. G. Gamow and M. Stern, Puzzle-Math, Viking, New York, 1958. January 2009] OVERHANG 43 6. M. Gardner, Mathematical games: Some paradoxes and puzzles involving infinite series and the concept of limit, Scientific American (Nov. 1964) 126–133. H. Freeman, 7. 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.

This observation enabled us to experiment with huge symmetric and asymmetric brickwall stacks. Figure 21. A schematic description of well-behaved collections of forces that balance symmetric and asymmetric brick-wall stacks. The best symmetric loaded brick-wall stacks with overhangs 10 and 50 that we have found are shown in Figures 22 and 23. 76 and 115,467, respectively. The blocks in the larger stack are so small that they are not shown individually. We again believe that these stacks are close to being the optimal stacks of their kind.

N. X i,n In particular, |Ui | ≤ 1 with probability one for all i = 2, . . , n, and therefore |X ∗In ,n | ≤ 1/σn → 0 (17) by (16). 2 may be invoked to show that the number of cycles of a random permutation is asymptotically normal. More generally, the small zero bias condition will hold for an array Xn with elements X i,n = X i /σn whenever the independent mean zero summand variables X 1 , X 2 , . . satisfy |X i | ≤ C with probability one for some constant C, and the variance σn2 of their sum Sn tends to infinity.

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American Mathematical Monthly, volume 116, number 1, january 2009 by Daniel J. Velleman


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