
D17. Transrotation
Posted: 2023 Nov 15 Solutions will be accepted through 2024 Mar 1Often you can slice up one shape and reassemble the pieces to make a different shape, like the famous squaretotriangle dissection shown in Figure 1. Notice that in this rearrangement you are forced to rotate some of the pieces when you put them back together. Sometimes, though, you can do the assembly by only translating…

D16. Five Choices
Posted: 2023 Nov 15 Solutions will be accepted through 2024 Mar 1You are at a crossroads with five roads leading away from it, and you don’t know which direction to go. You would like to choose randomly, with equal probability for each road. If your only way of making random choices is to flip a fair coin (one that could come up heads or tails with…

D15. Eccentric Repulsion
Posted: 2023 Nov 15 Solutions will be accepted through 2024 Mar 1Paul is back with another elliptical quandary. Suppose you are constrained to a circular track, which we will model as the unit circle in the plane, and there is something smelly at one point on the track, say where it intersects the x axis at the point ( 1 , 0 ) . If you…

D14. Flip Functions
Posted: 2023 Nov 15 Solutions will be accepted through 2024 Mar 1Call 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 01 Solutions will be accepted through 2024 Mar 1In 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…