• D7. Three and a Half Sides

    Three and a Half Sides

    Prof. Jim Propp of the University of Massachusetts Lowell has been corresponding with the Prisoner’s Dilemma about planar cross sections of three-dimensional solids. For example, suppose you have a regular tetrahedron (a triangular pyramid on an equilateral triangle base with the proper height so that all of the faces are equilateral triangles). If you pass…

    Posted: 2022 Sep 14
  • D6. Rectangle Pinwheel

    Rectangle Pinwheel

    The outermost polygon shown in figure 2 as the region shaded in pink consists of 16 congruent rectangles placed so that their edges align (and vertices of four of the rectangles coincide at the center of the polygon). We call this polygon a 28-gon because it has 28 edges: the line segments that make up…

    Posted: 2022 May 26
  • D5. Triple Fever

    Triple Fever

    Caught up in the excitement of the Derby and the Preakness, a buddy of yours heads to the track, and discovers that today is the running of the Medium-Rare Stakes. Only three horses have been entered: Tee Bone, New York Strip, and Rib Eye. Your buddy then notices the posted odds: Tee Bone is paying…

    Posted: 2022 May 23
  • D4. Four Operations Revisited

    Four Operations Revisited

    One of the great things about mathematics is that it’s always possible to look again at familiar things and discover something new. Sometimes this can happen when you combine well-known things in a new way. For example, you’re probably familiar with the four basic operations of addition, subtraction, multiplication, and division. And you may have…

    Posted: 2022 Jan 30
  • D3. Zero Sum Game

    Zero Sum Game

    Is it possible to number the edges of a cube using each of the numbers -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, and 6 once, so that for every vertex, the sum of the numbers on the edges that meet there is zero? (See figure 1 for a diagram of the…

    Posted: 2022 Jan 29
  • D2. Find Yourselves

    Find Yourselves

    In this classic dilemma, the director of a prison offers 100 people slated to be locked up one last chance at freedom. The individuals have been assigned different numbers from 1 to 100, and the director sets up a room containing 100 boxes. Each number from 1 to 100 has been put on a slip…

    Posted: 2022 Jan 28
  • D1. Continuing Irrationality

    Continuing Irrationality

    One interesting thing about continued fractions is that every infinite continued fraction represents an irrational number. But in our first dilemma, we ask what happens when you have an infinite sequence of finite continued fractions with more and more terms. Recall that the binomial coefficient ( a b ) has the formula a ! b…

    Posted: 2022 Jan 27

Archives: Dilemmas