Mathematical Puzzle: Funds of Sharing

This month we visit the trendy (but fictional) SP SS unusual feature: beds with hot mud.

Lay the plastic sheet on the mud. Then lay your body on a sheet. Without any direct contact between the mud and the body, spend a few minutes enjoying the soft and saunal heat, sweating throughout the plastic. Although the SPA session does not last long, it is said to be miraculously restorative.

One day three friends arrive. Unfortunately, only two plastic leaves are available. No one wants to miss; Then again, no one wants to lie on someone else’s sweat.

“Wait!” says one. “It’s easy! I’ll use one side of the sheet, and you can use the other.”

“Are you kidding?” Another answer. “That side will be covered with mud.”

The first friend smiles. “Not if we plan in advance.”

#1: How can all three friends participate in Toplice using only two leaves?

#2: The next day, five friends visited Toplice, with only three leaves available. Can I all participate? (Suppose the spa is now forbidden to set the already sensitive side of the sheet directly to their precious mud.)

#3: Soon 10 friends visit Toplica. Only five leaves are available. “Someone will have to miss,” one of them declares. “There’s no way to find out,” the other says, “until at least we look for a solution.” Who is right?

#4: Later, the spa introduces another type of mud, which should not be mixed with the first. If three friends want to try both muds, how many leaves need minimal? (Suppose each person is rude to lie twice on the same list.)

#5: Here, lurking is a complete general question, what mathematical researchers have yet to solve: what is the minimum number of leaves that allow n friends to experience m the type of mud if each side of the list can only touch one person or one type of mud? (You could start by assuming m = 1.)

As you try these puzzles, I recommend grabbing some index cards or sheets of manipulating paper. Or if you feel ambitious, grab some plastic leaves, some sweaty friends and a suitable mud patch.

Looking for answers? Go Sciencenews.org/puzzle-inswers. We would love to hear your thoughts. Send us e -a punetke@ciencenews.org.