A Painting Conundrum

painting conundrum

From Stephen Barr’s Experiments in Topology (1989) via Miodrag Petkovic’s Mathematics and Chess (1997):

This apartment contains eight rooms, each measuring 9 square meters, except for the top one, which measures 18 square meters. You have enough red paint to cover 27 square meters, enough yellow paint to cover 27 square meters, enough green paint to cover 18 square meters, and enough blue paint to cover 9 square meters. Can you paint the eight floors in four colors so that each room neighbors rooms of the other three colors?

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Equal Opportunity

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Can two dice be weighted so that the probability of each of the numbers 2, 3, …, 12 is the same?

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Half and Half

A bisecting arc is one that bisects the area of a given region. “What is the shortest bisecting arc of a circle?” Murray Klamkin asked D.J. Newman. Newman supposed that it was a diameter. “What is the shortest bisecting arc of a square?” Newman answered that it was an altitude through the center. Finally Klamkin asked, “And what is the shortest bisecting arc of an equilateral triangle?”

“By this time, Newman had suspected that I was setting him up (and I was) and almost was going to say the angle bisector,” Klamkin writes. “But he hesitated and said let me consider a chord parallel to the base and since this turns out to be shorter than an angle bisector, he gave this as his answer.”

Was he right?

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Change of Venue

A and B are playing a simple game. Between them are nine tiles numbered 1 to 9. They take tiles alternately from the pile, and the first to collect three tiles that sum to 15 wins the game. Does the first player have a winning strategy?

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The Wire Identification Problem

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A conduit carries 50 identical wires under a river, but their ends have not been labeled — you don’t know which ends on the west bank correspond to which on the east bank. To identify them, you can tie together the wires in pairs on the west bank, then row across the river and test the wires on the east bank to discover which pairs close a circuit and are thus connected.

Testing wires is easy, but rowing is hard. How can you plan the work to minimize your trips across the river?

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Sleeping Alone

Lori has an icky problem: Worms keep crawling onto her bed. She knows that worms can’t swim, so she puts each leg of the bed into a pail of water, but now the worms crawl up the walls of the room and drop onto her bed from the ceiling. She suspends a large canopy over the bed, but worms drop from the ceiling onto the canopy, creep over its edge to the underside, crawl over the bed, and drop.

Desperate, Lori installs a water-filled gutter around the perimeter of the canopy, but the worms drop from the ceiling onto the outer edge of the gutter, then crawl beneath. (The worms are very determined.) What can Lori do?

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