Category Archives: Puzzles and Problems

Spot in a Rectangle  Problem

This puzzle is from the Irishman Owen O’Shea.

“The following puzzle illustrates a beautiful mathematical relationship involving a rectangle of any size and a random point within that rectangle that most people, including mathematicians, are unaware of.

The figure shows a rectangular room.  There is a matchbox located 6 feet from one corner of the room and 27 feet from the opposite corner.  The matchbox is also located 21 feet from a third corner.

How far is the matchbox from the fourth corner?”

Answer.

See the Spot in a Rectangle Problem for a solution.

Yet Another Track Puzzle

This is another problem from Dan Griller.

“When Anthony and Benjamin run round a circular track in the same direction at constant but different speeds, they meet every 3 minutes.  When Benjamin changes direction (but maintains his speed) they meet every 40 seconds.

If Anthony is faster than Benjamin, calculate

(speed of Anthony) / (speed of Benjamin)”

Answer.

See Yet Another Track Puzzle for a solution.

Mixed Emotions

This is a brainteaser by S. Ageyev from the November-December 1991 issue of Quantum given in Futility Closet.

“The numbers 1, 2, …, 100 are arranged in a 10 x 10 square table in their natural order (1 in the top left comer, 100 in the bottom right comer). The signs of 50 of these numbers are changed in such a way that exactly half of the numbers in each line and each column get the minus sign. Prove that the sum of all the numbers in the table after this change is zero.”

See Mixed Emotions for solutions.

Three Circles Problem

This is a problem from the 1995 AIME problems.

“Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.”

Answer.

See the Three Circles Problem for solutions.

Hangover Clock Reading

This is another clock puzzle from the 1978 Eureka magazine.

“The hands on my alarm clock are indistinguishable, and there are no numbers around the outside. Accidentally woken up by it one morning, I observed with a snarl that the hands were both pointing at minute divisions, and that they were 9 minutes apart.

Had it not been for my hangover, what could I have deduced?”

Answer.

See the Hangover Clock Reading for a solution.

Functional Equation Puzzle

This is a math Olympiad problem from Puzzle Sphere where Muhammad Zain Sarwar claims it is at Harvard entrance exam level.

“Given the functional relationship  f(x + y) = f(x) + f(y) + xy with the known value  f(4) = 10, determine the value of f(2023).”

Just try some examples and detect the pattern that defines the function.

Answer.

See the Functional Equation Puzzle for a solution.

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