A problem from the 2011 Moscow Mathematical Olympiad: In a certain square matrix, the sum of the two largest numbers in each row is r and the sum of the two largest in each column is c. Show that r = c.
Puzzles
An “Excentrical Query”
Backward Baseball
In a retrograde analysis puzzle, one tries to deduce the history of a game from the current state of play. The most familiar examples concern chess, but Smith College mathematician Jim Henle worked out that it can also be done in baseball. This is the batting order of the Mudville Slugs:
- Flynn
- Blake
- Casey
- Hobbes
- Davis
- Shlabotnick
- Thayer
- Cooney
- Barrows
We’re told also that in the ninth inning Casey came to bat for the fourth time, while the bases were loaded with two men out. Casey struck out, leaving the team with another loss. How many runs did Mudville score altogether?
The Puzzle Wall
“Suppose there was a pond, around which four poor men built their houses, thus:”
“Suppose four evil-disposed rich men afterwards built houses around the poor people, thus:”
“… and wished to have all the water of the pond to themselves. How could they build a high wall, so as to shut out the poor people from the pond?”
Black and White
Weight Limit
In a set of weights, no weight exceeds 10 kg. If the set is divided arbitrarily into two groups, the combined mass of one of these groups also will not exceed 10 kg. What’s the greatest possible mass of the full set of weights?
Endangered Species
Kevin Purbhoo invented this vivid puzzle while a student at Northern Secondary School in Toronto:
On a remote Norwegian mountain top, there is a huge checkerboard, 1000 squares wide and 1000 squares long, surrounded by steep cliffs to the north, south, east, and west. Each square is marked with an arrow pointing in one of the eight compass directions, so (with the possible exception of some squares on the edges) each square has an arrow pointing to one of its eight nearest neighbors. The arrows on squares sharing an edge differ by at most 45 degrees. A lemming is placed randomly on one of the squares, and it jumps from square to square following the arrows. Prove that the poor creature will eventually plunge from a cliff to its death.
Round Trip
An interesting query by Bob High, posed in the May-June 1994 issue of MIT Technology Review: Suppose a billiard ball with a small black dot precisely on its top is rolled around the full circumference of a circle of the same radius. Assuming no slippage or twisting, where is the dot when the ball returns to its starting point?
Black and White
Unorthodox Behavior
A puzzle from the 18th century — punctuate this sentence so that it makes sense:
King Charles the First walked and talked half an hour after his head was cut off.