This is another train puzzle from H. E. Dudeney, which is fairly straight-forward.
“I put this little question to a stationmaster, and his correct answer was so prompt that I am convinced there is no necessity to seek talented railway officials in America or elsewhere. Two trains start at the same time, one from London to Liverpool, the other from Liverpool to London. If they arrive at their destinations one hour and four hours respectively after passing one another, how much faster is one train running than the other?”
See Two Trains – London to Liverpool
The Columbus story shows the intervention of chance in history at its most capricious. The following tale has its own logic, but the confluence of serendipitous events makes it marvelous and uplifting, especially in our current dark times. It was first brought to my attention by my father back in the early 1960s at the height of America’s role as wheat breadbasket of the world. America, and especially Kansas, was supplying essential wheat to the recently independent country of India and to the Soviet Union, whose long struggle with collective farming (and other factors), especially in the Ukraine, had led to its dependency on imports.
I will not try to narrate the story O’Henry-like with a surprise ending, but announce the amazing coincidence from the start—America was supplying the USSR its own wheat! The Kansas wheat was derived from a special hardy winter variety called Turkey Red that had originated in the Ukraine and was brought to America by Mennonites. So the story is how this all came about. See Turkey Red.
Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2009:
“Four positive integers a, b, c, and d are such that
_________abcd + abc + bcd + cda + dab + ab + bc + cd + da + ac + bd + a + b + c + d = 2009.
What is the value of a + b + c + d?
_________A 73_________B 75_________C 77_________D 79_________E 81”
See the Challenging Sum
(Update 4/17/2019) Continue reading
Since everyone by now who has any interest has gone directly to Catriona Shearer’s Twitter account for geometric puzzles, I was not going to include any more. But this one with its one-step solution is too fine to ignore and belongs with the “5 Problem” as one of the most elegant.
“Two squares sit on the hypotenuse of a right-angled triangle. What’s the angle?”
See the Two Block Incline Puzzle
(Update 4/26/2019) Continue reading
This is a challenging problem from Mathematical Quickies (1967).
“Evaluate the infinite product:”
I came up with a motivated solution using some standard techniques from calculus. Mathematical Quickies had a solution that did not employ calculus, but one which I felt used unmotivated tricks. See the Infinite Product Problem.
I came across this problem in Alfred Posamentier’s book, but I remember I had seen it a couple of places before and had never thought to solve it. At first, it seems like magic.
In any convex quadrilateral (line between any two points in the quadrilateral lies entirely inside the quadrilateral) inscribe a second convex quadrilateral with its vertices on the midpoints of the sides of the first quadrilateral. Show that the inscribed quadrilateral must be a parallelogram.
See the Magic Parallelogram.
(Update 5/15/2020) Continue reading
Here is a problem from the famous (infamous?) Putnam exam, presented by Presh Talwalkar. Needless to say, I did not solve it in 30 minutes—but at least I solved it (after making a blizzard of arithmetic and trigonometric errors).
“Today’s problem is from the 1978 test, problem B1 (the easiest of the second set of problems). A convex octagon inscribed in a circle has four consecutive sides of length 3 and four consecutive sides of length 2. Find the area of the octagon.”
My solution is horribly pedestrian and fraught with numerous chances for arithmetic mistakes to derail it, which happened in spades. As I suspected, there was an elegant, “easy” solution (as demonstrated by Talwalkar)—once you thought of it! Again, this is like a Coffin Problem. See the Putnam Octagon Problem.
This is one of H. E. Dudeney’s train puzzles.
“Two railway trains, one four hundred feet long and the other two hundred feet long, ran on parallel rails. It was found that when they went in opposite directions they passed each other in five seconds, but when they ran in the same direction the faster train would pass the other in fifteen seconds. A curious passenger worked out from these facts the rate per hour at which each train ran. Can the reader discover the correct answer? Of course, each train ran with a uniform velocity.”
See Two Trains – Passing in the Night.
It was reading Peter Hoffmann’s 2012 book Life’s Ratchet that drove home the role of determinism in biological processes, which he characterizes as a ratchet, a process that filters random behavior into a particular “purposeful” direction. Since Hoffmann is a biophysicist, his presentation is heavily guided by the physical principles of energy conversion, thermodynamics, and entropy, which makes for a fresh approach to a traditionally biological subject. The startling thing Hoffmann’s book introduced me to was the subject of molecular machines or motor proteins. These were amazing engines that harnessed the chemical and physical energy within a cell to act like miniature workers hauling materials around and constructing other molecules. The intelligent design crowd would go bonkers. See Chemical Determinism – Motor Proteins