# Octagonal Area Problem

Here is another problem from the Polish Mathematical Olympiads published in 1960.

“95. In a parallelogram of given area S each vertex has been connected with the mid-points of the opposite two sides.  In this manner the parallelogram has been cut into parts, one of them being an octagon.  Find the area of that octagon.”

See the Octagonal Area Problem for solutions.

# Another Christmas Tree Puzzle

This is a belated Christmas puzzle from December 2019 MathsMonday.

“A Christmas tree is made by stacking successively smaller cones. The largest cone has a base of radius 1 unit and a height of 2 units. Each smaller cone has a radius 3/4 of the previous cone and a height 3/4 of the previous cone. Its base overlaps the previous cone, sitting at a height 3/4 of the way up the previous cone.

What are the dimensions of the smallest cone, by volume, that will contain the whole tree for any number of cones?”

Recall that the volume of a cone is π r2 h/3.

See Another Christmas Tree Puzzle for a solution.

# Winter Sum

Here is another sum problem, this time from the 2021 Math Calendar.

________________

As before, recall that all the answers are integer days of the month.  And the solution employs a technique familiar to these pages.

See the Winter Sum for a solution.

# Grandfather Clock Puzzle

This is another doable puzzle from Sam Loyd.

“BACK OF THE OLDTIME song of “Grandfather’s clock was too tall for the shelf, so it stood for ninety years on the floor,” there was a legend of a pestiferous grand-father and a cantankerous old clock which, from the fitful time when “it was bought on the morn, when the old man was born,” it had made his whole life miserable, owing to an incurable habit which the clock had acquired of getting the hands tangled up whenever they attempted to pass.

These semi-occasional stoppages became of more frequent occurrence as advancing age made the old gentleman more irritable and his feeble hands more incapable of correcting the cranky antics of the balky old timepiece.

Once when the hands came together again and stopped the clock the old man flew into such an ungovernable passion that he fell down in a fit, stone dead, and it was then that

“The clock stopped short,

Never to go again,

When the old man died.”

A photograph of the clock was presented to me, showing the classical figure of a female representing time, and it struck me as remarkable that with the knowledge of the hour and minute hands being together that it should be possible to figure out the exact time at which “the old man died,” from the position of the second hand as shown, without having to see the face of the clock. The idea of being able to figure out the exact time of day from seeing the second hand alone is very odd, although not so difficult a puzzle as one would imagine.”

See the Grandfather Clock Puzzle for a solution.

# Circle Chord Problem

This is another nice puzzle from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008.

“The triangle ABC is inscribed in a circle of radius 1. Show that the length of the side AB is given by 2 sin c°, where c° is the size of the interior angle of the triangle at C.”

The diagram shows the case where C is on the same side of the chord AB as the center of the circle.  There is a second case to consider where C is on the other side of the chord from the center.

See the Circle Chord Problem

# Three Year Anniversary

Yet another year has passed, with dimming hopes for a return to “normal” from the pandemic.  Newton’s great achievements occurred during a similar time, so maybe something positive will arise out of our current difficulties.  As always, I hope things mathematical have provided a distraction and entertainment—and possibly even enrichment.

Again, I thought I would present the statistical pattern of interaction with the website in the absence of any explicit feedback.  I can’t draw any firm conclusions other than interest in the website seems to have reached a permanent plateau, and possibly a point of diminishing interest.  A cursory survey suggests that such websites as this have a half-life of about 3 years, so maybe it is a portent.

Anyway, here is the summary.

# River Traffic Problem

Here is another elegant Quantum math magazine Brainteaser problem.

“A raft and a motorboat set out downstream from a point A on the riverbank. At the same moment a second motorboat of the same type sets out from point B to meet them. When the first motorboat arrives at B, will the raft (floating with the current) be closer to point A or to the second motorboat? (G. Galperin)”

See the River Traffic Problem for solutions.

# Remainder Problem

Here is a challenging problem from the 2021 Math Calendar.

“Find the remainder from dividing the polynomial

x20 + x15 + x10 + x5 + x + 1

by the polynomial

x4 + x3 + x2 + x + 1”

Recall that all the answers are integer days of the month.

See the Remainder Problem for a solution.

# Ubiquitous Radius Problem

This is an interesting problem from Posamentier and Lehmann’s Mathematical Curiosities.

“In the figure we have a semicircle with the point P randomly placed on the diameter. Points A and B are situated on the circle such that they form angles of 60° with the diameter as shown in the figure. This problem asks us to show that the length of AB is equal to the radius of the semicircle.”

See the Ubiquitous Radius Problem

# Mystery of the Dancing Men

Manmohan Kaur took Arthur Conan Doyle’s popular 1903 Sherlock Holmes story “The Adventure of the Dancing Men” and used it in his math classes to illustrate the logic and mathematics involved in solving codes and ciphers.  I thought his idea might work as a puzzle.  It has been years since I read the story, so I had forgotten the decryption and found it quite doable from the setup provided by Kaur.  Here is his presentation, subject to further edits and reductions in size on my part.

“The original story has been shortened and simplified. Reference to England has been completely removed and some other superfluous information that distracts the reader instead of helping solve the mystery have been omitted. In the original story Elriges is the name of an inn but we have taken the liberty to use it loosely as the name of a town.

The pictures of all stick figure messages except the fourth are from the collection The Return of Sherlock Holmes. The original story has a typographical error that throws off the decryption scheme. To remove this (intentional or unintentional) error, the fourth figure has been taken from Trap and Washington’s Introduction to Cryptogra­phy with Coding Theory. The fourth message is meant to have a different handwriting, so this serves our purposes well.

Condensed Story

Hilton Cubitt of Elriges visits you and gives you a paper with the following mysterious sequence of stick figures that he found lying on the sun-dial in his mansion.

Message 1:

Cubitt explains that he recently married a Chicago woman named Elsie Patrick. Before the wedding, she had asked him never to ask about her past, as she had had some “very disagreeable associations” in her life, although she said that there was nothing that she was personally ashamed of. Their mar­riage had been a happy one until the messages began to arrive, first mailed from Chicago and then appearing in the garden of his mansion.

The messages had made Elsie very afraid but she did not explain the reasons for her fear, and Cubitt insisted on honoring his promise not to ask about Elsie’s life in Chicago. You look at the figures closely to understand them a little better and notice that some of the figures are holding flags. What could the flags mean? Perhaps the end of words?

The next morning Cubitt finds “a fresh crop of dancing men drawn in chalk upon the black wooden door of the tool-house”:

Message 2:

Two mornings later, “a fresh inscription had appeared”:

Message 3:

Three days later, “a message was left scrawled upon paper, and placed under a pebble upon the sun-dial”:

Message 4:

Cubitt gives copies of all these messages to you. Your task is to help him understand what is going on. You call your friend in the Chicago Police Department and ask her to find background information on Elsie Patrick. You learn that Elsie is the daughter of a Chicago crime boss, and was engaged to Abe Slaney, who worked for her dad, and that she had fled to escape her old life.

You examine all the occurrences of the dancing figures. Message 4 is in a different handwriting, so you guess that it is from a different person, most likely, Elsie, while messages 1, 2 and 3 are from the unknown person (the criminal). You spend the next two days trying to make some sense of the stick figures. You are now sure that the flags on some of the figures indicate the end of words. You also know that a simple substitution cipher is being used for the encryption, and that frequency analysis is the way to solve these ciphers.

Three days later, another message appears.

Message 5:

This message causes you to fear that the Cubitts are in immediate danger. You rush to Elriges and find Cubitt dead of a bullet to the heart and his wife gravely wounded from a gunshot to the head. What do the messages say?

Inspector Martin of the Norfolk Constabulary believes that it is a murder-suicide attempt; Elsie is the prime suspect. But you, after noting some inconsistencies in that theory, know that there is a third person involved. How will you prove to Inspector Martin that a third person is involved?”