
D10. Halfandhalf Hanoi
Posted: 2023 May 01You may be familiar with the Tower of Hanoi puzzle in which you start with eight discs stacked on one of three pegs, each disc smaller than the one below it. The other two pegs are empty. The goal of the puzzle is to move the entire stack of discs to a different peg, according…

D9. Semi Inellipse
Posted: 2022 Sep 15PMP participant Paul Morton has also investigated the properties of ellipses. As shown in Figure 1, semiellipse C T I is inscribed in right triangle C B A so that it is tangent to hypotenuse A B at T , and so that points I and the two foci F and G of the ellipse…

D8. Mullipse?
Posted: 2022 Sep 15Recall that given two points in the plane, an ellipse can be defined as the collection of points whose sum of distances to the given points is constant. But here in the Prisoner’s Dilemma, we know there are more operations than just addition. (Remember Dilemma 4?) So suppose you are given points F and G…

D7. Three and a Half Sides
Posted: 2022 Sep 14Prof. Jim Propp of the University of Massachusetts Lowell has been corresponding with the Prisoner’s Dilemma about planar cross sections of threedimensional 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…

D6. Rectangle Pinwheel
Posted: 2022 May 26The 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 28gon because it has 28 edges: the line segments that make up…

D5. Triple Fever
Posted: 2022 May 23Caught 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 MediumRare 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…

D4. Four Operations Revisited
Posted: 2022 Jan 30One 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 wellknown 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…

D3. Zero Sum Game
Posted: 2022 Jan 29Is 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…

D2. Find Yourselves
Posted: 2022 Jan 28In 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…

D1. Continuing Irrationality
Posted: 2022 Jan 27One 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…