
D14. Flip Functions
Posted: 2023 Nov 15Call a function f mapping the real numbers to themselves a signflipping function if for all x , y ∈ R , ( x − y ) ( f ( x ) + f ( y ) ) = ( x + y ) f ( x − y ) . For example, the identity…

D13. The Experbola
Posted: 2023 May 01In Dilemma 8, we investigated a curve similar to an ellipse, but changing addition to multiplication in its definition. This time, PMP participant William Keehn proposes a similar exploration related to the hyperbola, another conic section, which can be thought of as all of the points ( x , y ) in the plane where…

D12. Strange Coincidence
Posted: 2023 May 01Let’s listen in on a conversation between siblings divided by math, so to speak: “That’s amazing!” said Mathophila. “The squares of three consecutive positive whole numbers add up to the same number as the squares of the next two numbers.” “Oh, no big deal, that must happen all the time,” replied her brother, Innumeratus. “I…

D11. Octosection
Posted: 2023 May 01Square A B C D in Figure 1 has been dissected into eight equalarea rectangles. The width of the rectangle shaded red in Figure 1 is 35 units. What is the area of square A B C D ? For a bonus challenge, can you devise and solve a similar problem that dissects a square…

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…