Shuffle five pairs of socks, and randomly give two socks each to five of your friends. What is the probability that nobody gets a matching pair? That's the familiar problem of derangements with a new twist, and it's just one of the mysteries that are resolved in our April issue. Other mysteries involve doodles and knots, statistics and physics, and the game of bocce. —*Walter Stromquist*

Vol. 86, No. 2, pp.160-239.

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ARTICLES

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From Doodles to Diagrams to Knots

Colin Adams, Nöel MacNaughton, and Charmaine Sia

What closed curves can be drawn in the plane such that they cut the plane into complementary regions that are $$n$$-gons including the outer region, where $$n$$ is allowed to take some finite number of values? A curve is an $$(a_{1},a_{2},\dots,a_{n})$$-curve if the number of edges for its complementary regions all lie in $$a_{1},a_{2},\dots,a_{n}$$. We show that there are infinitely many curves for $$(2,n)$$, where $$n$$ is any odd integer greater than 3, and for $$(3n)$$, for any $$n>3$$ relatively prime to 3. We also consider the implications for knot theory, showing that every knot has a (3, 4, 5)-diagram. We ask what values of $$(a_{1},a_{2},\dots,a_{n})$$ will generate diagrams for every knot.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.83

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Deranged Socks

Sally Cockburn and Joshua Lesperance

It is an elementary combinatorial problem to determine the number of ways $$n$$ people can each choose two gloves from a pile of $$n$$ distinct pairs of gloves, with nobody getting a matching pair. Change the gloves to socks (with right socks being indistinguishable from left socks), however, and the problem becomes surprisingly more difficult. We show how this problem can be solved using a wide range of discrete mathematics tools: the principle of inclusion-exclusion; partitions; cyclic permutations; recurrence relations; as well as both ordinary and exponential generating functions. We even draw on a result from complex analysis to show that the fraction of all sock distributions that are deranged in this sense converges to $$1/e$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.97

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From Bocce to Positivity: Some Probabilistic Linear Algebra

Kent E. Morrison

A question in geometric probability about the location of the balls in a game of bocce leads to related questions about the probability that a system of linear equations has a positive solution and the probability that a random zero-sum game favors the row player. Under reasonable assumptions, we are able to find these probabilities.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.110

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The Use of Statistics in Experimental Physics

Thomas J. Pfaff, Maksim Sipos, M. C. Sullivan, B. G. Thompson, and Max M. Tran

Most mathematicians are aware of the importance of statistics in biological sciences, business, and economics, but are less aware that statistics is used every day in experimental physics. This paper gives three interesting examples of how statistics plays a vital role in physics. These examples use the basic statistical tools of residuals analysis and goodness of fit.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.120

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NOTES

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Continuous Function That Is Differentiable Only at the Rationals

Mark Lynch

An explicit construction is given of a function that is continuous on an interval, and differentiable only at the rationals.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.132

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Geometry of Cubic Polynomials

Sam Northshield

Imagine a sphere with its equator inscribed in an equilateral triangle. This Saturn-like figure will help us understand from where Cardano's formula for finding the roots of a cubic polynomial $$p(z)$$ comes. It will also help us find a new proof of Marden's theorem, the surprising result that the roots of the derivative $$p'(z)$$ are the foci of the ellipse inscribed in and tangent to the midpoints of the triangle determined by the roots of the polynomial.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.136

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What Is Special about the Divisors of 12?

Sunil K. Chebolu and Michael Mayers

What is an interesting number theoretic characterization of the divisors of 12 among all positive integers? This paper will provide one answer in terms of modular multiplication tables. We will show that the multiplication table for the ring $$\mathbb{Z}_{n}[x_{1},x_{2},\dots, x_{m}]$$ has 1's only on the diagonal if and only if $$n$$ is a divisor of 12.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.143

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Proof Without Words: The Length of a Triangle Median via the Parallelogram Law

C. Peter Lawes

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.146

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PROBLEMS

Proposals 1916-1920

Quickies 1029 & 1030

Solutions 1891-1895

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.147

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REVIEWS

Who's #1?, lost mathematics, and the Common Core

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.86.2.155