The Prisoner's Dilemma

D22. Fill the Jar

You are making sugar cookies from a batch of dough in the following way: You take your dough and roll it into a perfect square of just the right thickness (you are a very precise baker). Then you cut out 1-inch radius, perfectly circular, cookies arranged in a square grid so that the cookies are…
Posted: 2024-09-10 15:13:36
D21. Light Chase

You have a puzzle toy which consists of a ring of lit buttons, each of which can glow red, yellow, or (bluish-)green, as illustrated in Figure 1. When you push a button, its color cycles in the order red → green → yellow → red. The catch is, the lights two positions away in each…
Posted: 2024-09-10 14:00:15
D20. Geometry Game

You and a buddy start with a rectangular cake, represented by rectangle A B C D in Figure 1. Then you get to pick any point E between C and D , and the cake is sliced with two straight cuts, from B to E and E to A . Then it’s your friend’s turn…
Posted: 2024-03-01 21:08:40
D19. Covid Conundrum

Suppose you stood on line as part of a group of 30 people waiting to receive a vaccine. After everyone finished, you were told that ten members of the group turned out to have cases of Covid. You never came down with the infection, so you couldn’t have been one of the ten. If you…
Posted: 2024-03-01 20:39:35
D18. Domino Design

In the arrangement of five dominoes shown in Figure 1, obeying the usual rule that adjacent squares have matching numbers, the two at the end total five spots, and so do the three in the middle. There are three other ways to achieve this, not counting left-right reflections as different. Can you find them?
Posted: 2024-03-01 20:36:30
D17. Transrotation

Often you can slice up one shape and reassemble the pieces to make a different shape, like the famous square-to-triangle 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…
Posted: 2023-11-15 20:11:34
D16. Five Choices

You 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…
Posted: 2023-11-15 19:58:40
D15. Eccentric Repulsion

Paul 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…
Posted: 2023-11-15 19:37:25
D14. Flip Functions

Call a function f mapping the real numbers to themselves a sign-flipping function if for all x , y ∈ R , ( x − y ) ( f ( x ) + f ( y ) ) = ( x + y ) f ( x − y ) . For example, the identity…
Posted: 2023-11-15 19:11:33
D13. The Experbola

In 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…
Posted: 2023-05-01 13:40:34