Author Archives: Jim Stevenson

Al the Chemist II

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

Al the Chemist I

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?”

Answer.

See Al the Chemist I for solution.

Handicap Racing

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?”

Answer.

See the Handicap Racing for solution.

Peirce’s Law

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 →

Milk Mixing Puzzle

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?”

Answer.

See the Milk Mixing Puzzle for solution.

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?”

Answer.

See the Ant Connection Problem for solution.

Putnam Ellipse Areas 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.”

Answer.

See the Putnam Ellipse Areas Problem for solution.

A New Day

One of the physics blogs I enjoy reading is by the mathematical physicist Peter Woit, called Not Even Wrong.  A recent post provided a tantalizing teaser:

“I want to [link to] an insightful explanation of the history of string theory, discussing the implications of how it was sold to the public. It’s by a wonderful young physicist I had never heard of before, Angela Collier. She has a Youtube channel, and her latest video is string theory lied to us and now science communication is hard.

… It’s as hilarious as it is brilliant, and you have to see for yourself.”

Collier delivered her talk lucidly and thoroughly—all while playing a frenetic video game!  She claimed she used the length of the game to time her talk.  Of course we can walk and talk, and ride bicycles and talk, but I have never seen anyone split their mental concentration between a fast-paced video game and an esoteric physics explanation of the history of string theory and supersymmetry—for over 50 minutes!  And there was something about her presentation that was completely captivating.  It was definitely a serious scientific talk, but the ludicrousness of the game-playing echoed how ridiculous the continued, misplaced fascination with string theory is.  Naturally I had to learn more about this provocative physicist.

See A New Day

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