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

By Daniel J. Velleman

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In particular, y = inf{x : s(x) = s} is finite, and, by definition of y and the continuity of s(x), sup s(x) = r < s. x≤y− Since U and Vn are independent P(y ≤ U + Vn ≤ y + 1|Vn ) = s(y − Vn ) for all n. (33) Therefore, on the one hand we have P(y ≤ U + Vn ≤ y + 1|Vn ≥ ) ≤ r for all n ∈ K, but by conditioning on Vn ≥ and its complement, using (33), (32), (31), and the fact that U is absolutely continuous, we obtain the contradiction lim inf P(y ≤ U + Vn ≤ y + 1) ≤ r p + s(1 − p) n→∞ < s = P(y ≤ U ≤ y + 1) = lim P(y ≤ U + Vn ≤ y + 1).

Note that xi+1 − 1 < yi ≤ xi − 1, where xi is the x-coordinate of the left edge of Bi . 32 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 116 Shield block Bi applies a downward force of wi+1 on Bi+1 . , at the left edge of Bi+1 . Block Bi also applies a downward force of u i+1 on Bi+1 at z i+1 , where yi ≤ z i+1 ≤ yi+1 + 1. Similarly, block Bi−1 applies a downward force of u i on Bi at z i . Finally a downward external force of vi is applied on the left edge of Bi . The goal of the shield blocks is to aggregate the forces that should be applied on the spine blocks and to replace them by a set of fewer integral forces that are to be applied on the shield blocks.

Clarendon, Oxford, 1907. 14. M. Paterson and U. Zwick, Overhang, in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06), Society for Industrial and Applied Mathematics, Philadelphia, 2006, 231–240. 15. M. Paterson, Y. Peres, M. Thorup, P. Winkler, and U. 0093. 16. J. B. Phear, Elementary Mechanics, MacMillan, Cambridge, 1850. 17. A. Schrijver, Theory of Linear and Integer Programming, John Wiley, New York, 1998. 18. R. T. Sharp, Problem 52, Pi Mu Epsilon Journal 1 (1953) 322.

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