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?
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?
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?
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?
Tim Krabbé calls this “one of the funniest chess problems I ever saw.” Its composer, M. Kirtley, won first prize with it in a Problemist tourney in 1986.
It’s a selfmate in 8, which means that White must force Black to checkmate him 8 moves, despite Black’s best efforts to avoid doing so.
The solution is a single line — all of Black’s moves are forced:
1. Nb1+ Kb3 2. Qd1+ Rc2 3. Bc1 axb6 4. Ra1 b5 5. Rh1 bxc4 6. Ke1 c3 7. Ng1 f3 8. Bf1 f2#
All of White’s pieces have returned to their starting squares!
A problem from the U.S.S.R. mathematical olympiad:
You’re given 13 gears. Each weighs an integral number of grams. Any 12 of them can be placed on a pan balance, with 6 in each pan, so that the scale is in equilibrium. Prove that all the gears must be of equal weight.
A puzzle by Pierre Berloquin:
In my house are a number of rooms. (A hall separated from the rest of the house by one or more doors counts as a room.) Each room has an even number of doors, including doors that lead outside. Is the total number of outside doors even or odd?
From The Booke of Meery Riddles, 1629:
A soldier that to Black-heath-field went,
Prayed an astronomer of his judgment,
Which wrote these words to him plainly,–
Thou shalt goe thither well and safely
And from thence come home alive againe
Never at that field shalt thou be slaine.
The soldier was slaine there at that field,
And yet the astronomer his promise held.
By Alexander Yarosh. The position above was reached in a legal game, except that one piece has been knocked off the board. What was it?
A problem from the 2004 Moscow Mathematical Olympiad:
An arithmetic progression consists of integers. The sum of its first n terms is a power of two. Prove that n is also a power of two.
My wife and I walk up an ascending escalator. I climb 20 steps and reach the top in 60 seconds. My wife climbs 16 steps and reaches the top in 72 seconds. If the escalator broke tomorrow, how many steps would we have to climb?
Only one of these statements is true. Which is it?
A. All of the below
B. None of the below
C. One of the above
D. All of the above
E. None of the above
F. None of the above
A conundrum by French puzzle maven Pierre Berloquin:
Caroline leaves town driving at a constant speed. After some time she passes a milestone displaying a two-digit number. An hour later she passes a milestone displaying the same two digits but in reverse order. In another hour she passes a third milestone with the same two digits (in some order) separated by a zero.
What is the speed of Caroline’s car?
1. A friend gives you a bottle that contains seven olives. Two of them are green and five are black. He bets that if you remove three olives at random from the bottle, they’ll include a green one. Should you take the bet?
2. Agnes has a tin of olives. It originally contained both black and green ones, but someone has been eating them, so she’s not sure of the colors of the 14 olives that remain. She removes 7 at random and finds that they’re all green. If the odds of this happening were exactly 50-50, what are the colors of the remaining 7?
In 2000, the residents of Luppitt, East Devon, installed a granite bench decorated with a variety of puzzles and curiosities that “it is hoped will be practical and entertaining for most of the next millennium.”
Among the puzzles is this “railway maze,” contributed by Roger Penrose. Make your way from Start in the upper left to Finish in the lower right. The catch is that your train has no reverse gear — you must move continually forward, following the natural curve of the track and making no sharp turns.
Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.
This sounds dead simple, but it stumped me for a while. See if you can do it. (There’s no trick — the task is just what it seems.)
In S.S. Van Dine’s The Bishop Murder Case (1929), someone is killing chessplayers and leaving a black bishop at each crime scene. The prime suspect is John Pardee, promoter of a chess opening called the Pardee Gambit, which he hopes to establish in master play. But Pardee kills himself, despondent after losing to Akiba Rubinstein at the Manhattan Chess Club. It turns out that the real killer was only using the chess angle to throw suspicion onto others.
Van Dine based Pardee on a real person, Isaac Leopold Rice, who sponsored numerous tournaments in which his Rice Gambit was the required opening. But practice showed that the best White could hope for was a draw, and the line was abandoned after World War I. In 1979 Larry Evans wrote, “One of the most heavily analyzed openings in history is now never played, interred in a footnote of the latest opening manual.”
In the book, investigators determine that Pardee had faced the position above against Rubinstein shortly before his suicide. White has just realized that Black has a forced win in four moves. How does Black play?