# Yet Another Race

This is another race puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings for 2013.

“In a 100 meter race, Jacob can beat Johann by 5 meters, and Johann can beat Nicolaus by 10 meters. By how much can Jacob beat Nicolaus?”

See Yet Another Race for a solution.

# Floating Square Puzzle

This is another puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“A mysterious square has materialized in the middle of the MCG, hovering in mid-air. The heights above the ground of three of its corners are 13, 21 and 34 metres. The fourth corner is higher still. How high?”

See the Floating Square Puzzle for solutions.

(Update 8/13/2023)  Alternative Solution Continue reading

# Ant Connection Problem

This is a nice puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“Two ants are on a cylindrical glass that is 5 centimetres in diameter. The ants are on opposite sides of the glass, 5 centimetres down from the glass’s rim. If both ants are on the outside of the glass, what is the shortest distance required for one ant to crawl to the other? What if one ant is on the outside of the glass and the other is on the inside?”

See the Ant Connection Problem for solution.

# Covering Rectangle Puzzle

This is a nice puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“In the picture does the green rectangle cover more or less than half of the [congruent] red rectangle?”

It is evident from the problem solution that the two rectangles are the same, so I made it explicit.

See the Covering Rectangle Puzzle for solutions.

# Spiral Areas Puzzle

This is a provocative puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings for 2013.

“In the picture the top curve is a semicircle and the bottom curve is a quarter circle. Which has greater area, the red square or the blue rectangle?”

See the Spiral Areas Puzzle for solutions.

# Clock Connections Puzzle

This is an imaginative puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings for 2012.

“You draw a line connecting the 5 and 9 on a clock face, and another line connecting the 3 and 8. What is the angle between the two lines?”

See the Clock Connections Puzzle for solutions.

This is a thoughtful puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“A ladder is leaning against a wall. The base of the ladder starts sliding away from the wall, with the top of the ladder sliding down the wall. As the ladder slides, you watch the red point in the middle of the ladder. What figure does the red point trace? What about other points on the ladder?”

See the Ladder Locus Puzzle for solutions.

# Existence Proofs

Here is a seemingly simple problem from Futility Closet.

“A quickie from Peter Winkler’s Mathematical Puzzles, 2021: Can West Virginia be inscribed in a square? That is, is it possible to draw some square each of whose four sides is tangent to this shape?”

Technically we might rephrase this as, can we inscribe a flat map of West Virginia in a square, since the boundary of most states is probably not differentiable everywhere, that is, has a tangent everywhere.

But the real significance of the problem is that it is an example of an “existence proof”, which in mathematics refers to a proof that asserts the existence of a solution to a problem, but does not (or cannot) produce the solution itself.  These proofs are second in delight only to the “impossible proofs” which prove that something is impossible, such as trisecting an angle solely with ruler and compass.

Here is another classic example (whose origin I don’t recall).  Consider the temperatures of the earth around the equator.  At any given instant of time there must be at least two antipodal points that have the same temperature.  (Antipodal points are the opposite ends of a diameter through the center of the earth.)

See Existence Proofs (revised)

(Update 10/2/2021) I fixed a minor typo: “tail” should have been “head”