When Bernoulli was preparing his *Ars Conjectandi*, it was hard for him to know what others had done even decades earlier. By 1876, when Charles Hugo Kummell began publishing in American math journals, there were few of them to choose from. Now we have a much more active mathematical community—and it is made real by the talented people listed in the back of this issue, in the Acknowledgments and Index sections.

There's a crossword puzzle! Turn to page 370, or download a copy from this page. —*Walter Stromquist*

Vol. 86, No. 5, pp.319-399.

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Gerald L. Alexanderson

An exposition of Jacques Bernoulli’s contribution to an oft-visited problem: finding formulas for the sum of the *k*th powers of the first *n* positive integers, formulas seen in beginning calculus courses. Bernoulli published his results in his masterpiece, *Ars Conjectandi*, published in 1713, a famous work which we celebrate in this, its tercentenary year.

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Asta Shomberg and James Tattersall

European statistical research flourished in the years 1770 to 1850 whereas statistical research in the United States did not develop fully until the latter half of the nineteenth century. The establishment of the United States Coast Survey, the Lake Survey, and the Nautical Almanac in 1807, 1841, and 1849, respectively, encouraged the rapid advancement of interest in statistics and the development of statistical methods. It was pioneered by Robert Adrain, Benjamin Peirce, his son Charles Sanders Peirce, Simon Newcomb, and Erastus Lyman De Forest, whose work is well researched. In this article we focus on life and the statistical accomplishments of Charles Hugo Kummell, a statistician for the Lake Survey and the U.S. Coast and Geodetic Survey, and an active member of the Philosophical Society of Washington. We describe his research into laws of errors of observations and his contributions to the development of the least-squares method.

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John H. Clifford and Michael Lachance

The singular value decomposition is a workhorse in many areas of applied mathematics and the insights it gives to linear transformations is beautiful. Using the geometry given by the SVD, we prove that the critical points of a quartic polynomial whose zeros are the vertices of a parallelogram are the foci and center of an inscribed ellipse passing through the midpoints of the sides of the parallelogram.

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Roger B. Nelsen

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Alessandro Fonda

We propose a general formula involving *n* points in a Euclidean space, which generalizes, on one hand, a well-known formula for the medians of a triangle and, on the other hand, two other formulas involving either the medians or the bimedians of a tetrahedron.

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Vincent J. Matsko

The astroid is well known as an envelope both of a family of line segments and a family of ellipses. The relationship between these two families is investigated by asking when two generic ellipses are tangent to each other, where a generic ellipse is described by $$|x/a|^{m}+|y/b|^{m}=1$$ for some $$a, b, m>0$$. Results are illustrated with several examples of envelopes of families of generic ellipses.

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Nicole Yunger Halpern

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Eli Dupree and Ben Mathes

We describe how to turn an enumeration of the rational numbers into a function with a dense graph.

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Brendan W. Sullivan

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Marc Renault

For given integers $$a$$ and $$b$$, we consider the $$(a,b)$$-Fibonacci sequence $$F$$ defined by $$F_{0} = 0$$, $$F_{1} = 1$$, and $$F_{n} = aF_{n-1}+bF_{n-2}$$. Given $$m\geq2$$ relatively prime to $$b$$,$$F(\bmod{m})$$ is periodic with period denoted $$\pi(m)$$. The rank of $$F(\bmod{m})$$, denoted $$\alpha(m)$$, is the least positive $$r$$ such that $$F_{r} = 0 (\bmod{m})$$, and the order of $$F(\bmod{m})$$, denoted $$\omega(m)$$, is $$\pi(m)/\alpha(m)$$. In this article, we pull together results on $$\pi(m)$$, $$\alpha(m)$$, and $$\omega(m)$$ from the classic case $$a=1$$, $$b=1$$, and generalize their proofs to accommodate arbitrary integers $$a$$ and $$b$$. Matrix methods are used extensively to provide elementary proofs.

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Proposals 1931-1935

Quickies 1035 & 1036

Solutions 1906-1910

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"Real" numbers, mathematical bull, interactive engagement, and the ultimate proof

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