Kepler’s second law holds that a line segment connecting an orbiting planet to its sun sweeps out equal areas in equal periods of time: In the diagram above, if the time intervals t are equal, then so are the areas A.
If gravity were turned off, would this still be true?
Imagine a 1000 x 1000 chessboard on which a white king and 499 black rooks are placed at random such that no rook threatens the king. And suppose the king goes bonkers and wants to kill himself. Can he reach a threatened square in a finite number of moves if Black is trying actively to avoid this?
The names of two U.S. state capitals end with the same eight letters. What are they?
Walter Penney of Greenbelt, Md., offered this poser in the August 1969 issue of Word Ways: The Journal of Recreational Linguistics. Below are five groups of English words. Each group appears also in a foreign language. What are the languages?
- aloud, angel, hark, inner, lover, room, taken, wig
- alas, atlas, into, manner, pore, tie, vain, valve
- ail, ballot, enter, four, lent, lit, mire, taller
- banjo, chosen, hippo, pure, same, share, tempo, tendon
- ago, cur, dare, fur, limes, mane, probe, undo
In projective geometry, every family of parallel straight lines intersects at an infinitely distant point. Chess problem composers in the former Yugoslavia have adapted this idea for the chessboard, adding four special squares “at infinity.”
Now a queen on a bare board, for example, can zoom off to the west (or east) and reach a square “at infinity” from which she attacks every rank on the board simultaneously from both directions. She might also zoom to the north (or south) to reach a different square at infinity; from this one she attacks every file simultaneously, again from both directions. Finally she can zoom to the northwest or southeast and attack all the diagonals parallel to a8-h1, or zoom to the northeast or southwest and attack all the diagonals parallel to a1-h8. These four “infinity squares,” plus the regular board, make up the field of play.
N. Petrovic created the problem below, published in Matematika Na Shahmatnoi Doske. White is to play and mate in at least two moves. Can you find the solution?
A Russian problem from the 1999 Mathematical Olympiad:
A father wants to take his two sons to visit their grandmother, who lives 33 kilometers away. His motorcycle will cover 25 kilometers per hour if he rides alone, but the speed drops to 20 kph if he carries one passenger, and he cannot carry two. Each brother walks at 5 kph. Can the three of them reach grandmother’s house in 3 hours?
From Sam Loyd:
Two children, who were all tangled up in their reckoning of the days of the week, paused on their way to school to straighten matters out.
“When the day after tomorrow is yesterday,” said Priscilla, “then ‘today’ will be as far from Sunday as that day was which was ‘today’ when the day before yesterday was tomorrow!”
On which day of the week did this puzzling prattle occur?
Six boys are accused of stealing apples. Exactly two are guilty. Which two? When the boys are questioned, Harry names Charlie and George, James names Donald and Tom, Donald names Tom and Charlie, George names Harry and Charlie, and Charlie names Donald and James. Tom can’t be found. Four of the boys who were questioned named one guilty boy correctly and one incorrectly, and the fifth lied outright. Who stole the apples?
A group of four missionaries are on one side of a river, and four cannibals are on the other side. The two groups would like to exchange places, but there’s only one rowboat, and it holds only three people, and only one missionary and one cannibal know how to row, and the cannibals will overpower the missionaries as soon as they outnumber them, either on land or in the boat. Can the crossing be accomplished?
A brainteaser by Chris Maslanka:
A packet of sugar retails for 90 cents. Each packet includes a voucher, and nine vouchers can be redeemed for a free packet. What is the value of the contents of one packet? (Ignore the cost of the packaging.)
A variation on yesterday’s puzzle:
Suppose there are six bottles of pills, and more than one of them may contain defective pills that weigh 6 grams instead of 5. How can we identify the bad bottles with a single weighing?
An efficiency-minded pharmacist has just received a shipment of 10 bottles of pills when the manufacturer calls to say that there’s been an error — nine of the bottles contain pills that weigh 5 grams apiece, which is correct, but the pills in the remaining bottle weigh 6 grams apiece. The pharmacist could find the bad batch by simply weighing one pill from each bottle, but he hits on a way to accomplish this with a single weighing. What does he do?
This week’s puzzle has a twist: Imagine that the board has been rolled into a cylinder so that the a- and h-files are joined and pieces can move across the boundary. How can White mate in two moves?
The first 10 letters of the alphabet, ABCDEFGHIJ, form a cipher that conceals the name of a number less than 100. What is the number?
- How does a deaf man indicate to a hardware clerk that he wants to buy a saw?
- How can you aim your car north on a straight road, drive for a hundred yards, and find yourself a hundred yards south of where you started?
- What runs fore to aft on one side of a ship and aft to fore on the other?
- A very fast train travels from City A to City B in an hour and a quarter. But the return trip, made under identical conditions, requires 75 minutes. Why?
- Does Canada have a 4th of July?
- Exhausted, you go to bed at 8 p.m., but you don’t want to miss an appointment at 10 a.m. the next day, so you set your alarm clock for 9. How many hours do you sleep?