Another puzzle by Presh Talwalkar.
“Thanks to John H. for the suggestion!
A square is inscribed in a quarter circle such that the outer vertices are on the arc of the quarter circle. If the quarter circle has a radius equal to 1, what is the area of the square?
I am told this was given to 7th grade students (ages 12-13), and I think it is a very challenging problem for that age group. In fact I think it is a good problem for any geometry student.”
See the Square in Quarter Circle for solutions.
A prevalent theme of much of popular mathematical exposition and debates about mathematics education concerns how to interest a wider population in matters mathematical. For the most part I feel that essays that try to present the “beauty” of mathematics are doomed to failure, as are most discussions of esthetics. The underlying goal of such writing is a legitimate and laudable attempt to show the appeal of math. But I fear it succeeds only with those already converted. So is there another way?
See the Lure of Mathematics Conundrum
After a hiatus of four years, Stephen Welch is back with some timely videos at Welch Labs that just coincidentally occur around the time of the new movie release of Oppenheimer. They deal with the history of the physics behind the Manhattan Project at Los Alamos. Continue reading
This is a problem from the 1987 American Invitational Mathematics Exam (AIME).
“Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al’s speed of walking (in steps per unit time) is three times Bob’s walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)”
See the Escalator Puzzle for solutions.
This is the second part of the problem from Raymond Smullyan in the “Brain Bogglers” section of the 1996 Discover magazine.
“On another occasion, Al made a mixture of water and wine. There was more water than wine—in fact, the excess of water over wine was equal to one-fourth the quantity of wine. Al then added 12 ounces of wine, at which point there was one ounce more of wine than water.
According to another version of the story, before Al added the 12 ounces of wine, he first boiled off 12 ounces of water (the net effect being that he replaced 12 ounces of water with wine), and again there was one more ounce of wine than water.
Would there be more mixture present at the end of the first version or the second?”
I found this statement a tad ambiguous with the result that I found two possible solutions: the one Smullyan gave and another, surprising one.
This is a relatively simple problem from the inventive Raymond Smullyan in the “Brain Bogglers” section of the 1996 Discover magazine.
“AL THE CHEMIST—not an alchemist, though his name might suggest it—one day partially filled a container with some concoction or other. He knew the volume of fluid in the container, as well as the volume of empty space, and realized that two-thirds of the former was equal to four-fifths of the latter. Was the container then less than half full, more than half full, or exactly half full?”
See Al the Chemist I for solution.
This is a nice variation on a racing problem by Geoffrey Mott-Smith from 1954.
“On one side of the playground some of the children were holding foot-races, under a supervisor who handicapped each child according to age and size. In one race, she placed the big boy at the starting line, the little boy a few paces in front of the line, and she gave the little girl twice as much headstart over the little boy as he had over the big boy. The big boy won the race nevertheless. He overtook the little boy in 6 seconds, and the little girl 4 seconds later.
Assuming that all three runners maintained a uniform speed, how long did it take the little boy to overtake the little girl?”
See the Handicap Racing for solution.
The June 2023 Carnival of Mathematics # 216 at Eddie’s Math and Calculator Blog has the rather arresting item concerning Peirce’s Law from the American logician Charles Sanders Peirce (1839 – 1914).
“Peirce’s Law: Jon Awbrey of the Inquiry Into Inquiry blog
This article explains Pierce’s Law and provides the proof of the law. The proof is provided in two ways: by reason and graphically. Simply put, for propositions P and Q, the law states:
P must be true if there exists Q such that the statement “if P then Q” is true. In symbols:
(( P ⇒ Q) ⇒ P) ⇒ P
The law is an interesting tongue twister to say the least.”
Perhaps another way of saying it is “if the implication P ⇒ Q implies that P is true, then P must be true.” Still, it sounds weird.
See Peirce’s Law
(Update 6/20/2023) Appendix: Valid Argument Continue reading
The Futility Closet website had the following problem:
“In isosceles triangle ABC, CD = AB and BE is perpendicular to AC. Show that CEB is a 3-4-5 right triangle.”
See a Triangle Puzzle
Here is another problem from the “Challenges” section of the Quantum magazine.
“Point L divides the diagonal AC of a square ABCD in the ratio 3:1, K is the midpoint of side AB. Prove that angle KLD is a right angle. (Y. Bogaturov)”
