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
Puzzles and Problems: AIME, travel puzzles
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.
See Al the Chemist II
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
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
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)”
See Right Angles in a Square
This is a classic example of a mixture problem from Dan Griller that recalls my agonies of beginning algebra.
“In Cauchy Village, full fat milk has 3.5% fat content, semi-skimmed milk has a 1.5% fat content, and skimmed milk has a 0.2% fat content. How many liters of full fat milk must be added to 100 liters of skimmed milk to produce semi-skimmed milk?”
See the Milk Mixing 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.
“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.
This is a nifty problem from Presh Talwakar.
“This is adapted from the 1994 Putnam, A2. Thanks to Nirman for the suggestion!
Let R be the region in the first quadrant bounded by the x-axis, the line y = x/2, and the ellipse x2/9 + y2 = 1. Let R‘ be the region in the first quadrant bounded by the y-axis, the line y = mx and the ellipse. Find the value of m such that R and R‘ have the same area.”
See the Putnam Ellipse Areas Problem