This is a most interesting problem proposed by Mirangu and retweeted by Catriona Agg:

“Two equilateral triangles share a vertex. What is the proportion red : green?”

See the Two Equilateral Triangles for solutions.

This is a most interesting problem proposed by Mirangu and retweeted by Catriona Agg:

“Two equilateral triangles share a vertex. What is the proportion red : green?”

See the Two Equilateral Triangles for solutions.

This is a thoughtful puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“A ladder is leaning against a wall. The base of the ladder starts sliding away from the wall, with the top of the ladder sliding down the wall. As the ladder slides, you watch the red point in the middle of the ladder. What figure does the red point trace? What about other points on the ladder?”

See the Ladder Locus Puzzle for solutions.

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.

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 r^{2 }h/3.

See Another Christmas Tree Puzzle for a solution.

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.

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.

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

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.

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.

Here is a challenging problem from the 2021 Math Calendar.

“Find the remainder from dividing the polynomial

*x*^{20} + *x*^{15} + *x*^{10} + *x*^{5} + *x* + 1

by the polynomial

*x*^{4} + *x*^{3} + *x*^{2} + *x* + 1”

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

See the Remainder Problem for a solution.