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College Mathematics Journal Contents—September 2017

This annual games and puzzles issue highlights squares: Iwan Praton considers the perimeter sum of squares tiling the unit square and Patrick Costello tiles squares with big holes using L-trominoes, while David Nacin not only provides three "dihedoku" puzzles but contributes an article about complex KenKen.

Even the Carcassonne game that Mindy Capaldi and Tiffany Kolba discuss using in the probability classroom is played with square cards. The pattern is broken with the rectangular SET cards in Jonathan Needleman and Felicia Sciortino's article, but what's a September CMJ issue without a new SET structure?

And rather than review one book, Jason Rosenhouse surveys the collected work of the recently deceased Raymond Smullyan, well known for his recreational logic writings. —Brian Hopkins

Vol. 48, No. 4, pp. 241-320


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Minimal Tilings of a Unit Square

p. 242.

Iwan Praton

Tile the unit square with a fixed number of small squares. We determine the minimum of the sum of the side lengths of the small squares, where the minimum is taken over all possible ways to tile the unit square with the same number of squares.

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Dihedoku Puzzle 1

p. 248.

David Nacin

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UFOs in the game SET: Looking for Airplanes and Spaceships

p. 249.

Jonathan Needleman and Felicia Sciortino

The mathematics of the popular card game SET has been extensively studied. Recently, planets were introduced as two-dimensional sets. We take this one step further by introducing spaceships, a three-dimensional set. We study the properties of spaceships along with the related concepts of UFOs and airplanes in SET. Finally, we determine how many red cards are needed to guarantee the existence of a spaceship.

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Dihedoku Puzzle 2

p. 258.

David Nacin

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Tiling Squares with Big Holes with L-trominoes

p. 259.

Patrick J. Costello

Consider a square board whose length is a power of two with a square consisting of one fourth of its area removed. We show that some of these “severely deficient” can be tiled by L-trominoes, depending on the position of the missing squares, and ask about other positions.

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Dihedoku Puzzle 3

p. 264.

David Nacin

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Carcassonne in the Classroom

p. 265.

Mindy Capaldi and Tiffany Kolba

Learning about probability can be hard and frustrating for many students. However, learning about probability through examples with board games can make this task more interesting and fun. We present a sequence of increasingly difficult probability problems derived from the popular board game Carcassonne. Each question is appropriate either for a college classroom or for undergraduate research, with topics including basic counting problems, expected value, Bayes’s theorem, Markov chains, and Monte Carlo simulation. Some problems have solutions, but other questions are left open for the reader to explore.

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On a Complex KenKen Problem

p. 274.

David Nacin

In his popular blog, Brian Hayes presented a four-by-four KenKen puzzle over a set of complex numbers, mentioning that he had not checked the uniqueness of the solution. Here, we study all cage patterns over similar puzzles that allow for unique solutions and show that they are equivalent to Hayes’s choice. We define a group action that preserves the size of solution sets and proceed to classify all puzzles with unique solutions before concluding with suggested activities and several open questions.

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Dihedoku Puzzles Solutions

p. 283.

David Nacin

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Classroom Capsules

The Demise of Trig Substitutions?

p. 284.

David Betounes and Mylan Redfern

After decades of using trigonometric substitutions to compute certain types of integrals, we were led to an easy alternative to trigonometric substitutions while grading student work. These “u-square” substitutions may be the answer to every student’s dream: No trigonometric functions are required (except sometimes the inverse tangent).

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A Short Proof of the Bolzano-Weierstrass Theorem

p. 288.

Katrina Eidolon and Greg Oman

We present a short proof of the Bolzano–Weierstrass theorem on the real line which avoids monotonic subsequences, Cantor’s intersection theorem, and the Heine–Borel theorem.

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Relaxing the Integral Test: A Challenge for the Advanced Calculus Student

p. 290.

Paul Carter and Yitzchak Elchanan Solomon

Illustrative, elementary counterexamples are hard to come by. In this note we propose an elementary, closed-form counterexample to a generalization of the classic integral test where the condition of monotonicity is relaxed. The analysis only uses techniques accessible to a second-semester calculus student.

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Problems and Solutions

p. 292.

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Book Reviews

The Works of Raymond Smullyan

p. 302.

Surveyed by Jason Rosenhouse

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Media Highlights

p. 313.

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