Kissing Angles

I really was trying to stop including Catriona Shearer’s problems, since they are probably all well-known and popular by now. But this is another virtually one-step-solution problem that again seems impossible at first. Many of her problems entail more steps, but I am especially intrigued by the one-step problems.

“What’s the sum of the two marked angles?”

See Kissing Angles for a solution.

An amazing publication was conceived primarily for women at the beginning of the 18th century in 1704 and was called The Ladies’ Diary or Woman’s Almanack. What made it even more remarkable was that each issue contained mathematical problems whose solutions from the readers were provided in the next issue. One particularly sharp woman was Mary Wright (Mrs. Mary Nelson). This is one of her problems:

VIII. Question 72 by Mrs. Mary Nelson
(proposed in 1719, answered in 1720)

A prize was divided by a captain among his crew in the following manner: the first took 1 pound and one hundredth part of the remainder; the second 2 pounds and one hundredth part of the remainder; the third 3 pounds and one hundredth part of the remainder; and they proceeded in this manner to the last, who took all that was left, and it was then found that the prize had by this means been equally divided amongst the crew. Now if the number of men of which the crew consisted be added to the number of pounds in each share, the square of that sum will be four times the number of pounds in the chest: How many men did the crew consist of, and what was each share?”

What makes this problem nice is that it does have a clean answer, contrary to most of the problems in The Ladies’ Diary.

See the Ladies’ Diary Problem for solutions.

Two Trains – London to Liverpool

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 for a solution.

Turkey Red

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.

(Update 2/24/2022)  Russian Invasion of Ukraine

More ironic coincidences.  Who would have thought a story about wheat from a distant land over one hundred years ago would become timely in the 21st century.  But terms like Catherine the Great’s “New Russia” are being uttered by a modern despot Vladimir Putin and port cities like Mariupol are again in the news.  All attention has been on focused on the effect of Putin’s folly on oil and gas, but I have been wondering about its consequences for the wheat.  And sure enough, such concerns are finally in the news.

See the Russian Invasion.

(Update 2/28/2022)  Russia May Weaponize Food Supply Chain

Politico has a more expansive article on the implications of the grain production in Ukraine by Ian Ralby et al., “Why the U.S. Needs to Act Fast to Prevent Russia from Weaponizing Food Supply Chains”.  For example, the article asserts, “Amid the chaos of this conflict and the threat to Ukrainian lives and independence, one critical implication has been grossly underexamined: how Russia could rely on China’s support to weaponize global food supply chains.”  Though Russia’s gas and oil production has garnered the most attention, “Russia’s control of Ukrainian grain shipments will likely have far greater consequences.  After just one day of the invasion, Russia effectively controlled nearly a third of the world’s wheat exports, three quarters of the world’s sunflower oil exports, and substantial amounts of barley, soy and other grain supply chains.”  The article examines in detail the implications of this control.

Challenging Sum

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 for a solution.

Two Block Incline Puzzle

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 for a solution.

Infinite Product Problem

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 for solutions.

Magic Parallelogram

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.

Putnam Octagon Problem

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 for solutions.

Two Trains – Passing in the Night

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 for a solution.