Many middle school students approach math with caution, often trepidation, and generally less-than-complete confidence. Several factors contribute to their views of what math actually is (a system of rules to follow and formulas to apply) as well as who might be good at it (people who can see the path to an answer and calculate quickly). A few say…

The author provides an example of obtaining the correct answer to a problem through a so-called nonsense solution. The conditions under which this incorrect method will yield a correct solution are examined. (JP)

Elucidated is the relationship among three threads of mathematical investigations: Kirkman's schoolgirl problems, finite geometries, and Euler's n-square officer problems. (JP)

Examples of logical paradoxes, including the square root of two paradox, Achilles and the tortoise paradox, author paradox, Russell paradox, bibliomaniac paradox, and Berry paradox, are presented; some are resolved. (DT)

Illustrated is the mathematical analysis of a three-dimensional game using techniques from the calculus of finite differences. This game is an analog of the familiar two-dimensional game, Dots and Squares. (JP)

The process that Hippasus of Metapontum (ca. 470 B.C.) may have used to discover irrational numbers is reconstructed. (MP)

An approach to how to expand explorations of determinants is detailed that allows evaluation of the fourth order. The method is built from a close examination of the product terms found in the expansions of second- and third-order determinants. Students are provided with an experience in basic mathematical investigation. (MP)

Results are presented of an impromptu exploration of polar formulas for volumes of revolution for certain plane regions. The material is thought to be unique, and to offer room for student exploration. It is felt pupil investigation can lead to increased pupil interest in both polar coordinates and calculus. (MP)

The question is asked, "How many gifts in total were given in 'The Twelve Days of Christmas' song," and is shown to relate to several other problems. Most of the number patterns and partitions dealt with are viewed to be usable with students in junior and senior high school. (MP)

The beauty of discovering some simple yet elegant proof either to something new or to an already established fact is discussed. A combinatorial problem that deals with covering a checkerboard with dominoes is presented as a starting point for individual investigation of similar problems. (MP)

A hungry worm is looking for something to eat according to very specific rules, and the path he takes is a graph. The problem is detailed in Applesoft BASIC using low resolution graphics for worms that turn 90 degrees and high resolution for worms that can turn 45 degrees. (MP)

Some patterns in prime numbers and their implications for general theorems are presented. Much of the material is taken from "The Strong Law of Small Numbers," an unpublished paper by Richard Kenneth Guy. (MP)