This is a nice little puzzle from the 2024 Math Calendar.

“Find the sum of the coefficients of

(1 + *x + x*^{2})^{3 }“

As before, recall that all the answers are integer days of the month.

See the Simple Polynomial Puzzle for a solution.

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This is a nice little puzzle from the 2024 Math Calendar.

“Find the sum of the coefficients of

(1 + *x + x*^{2})^{3 }“

As before, recall that all the answers are integer days of the month.

See the Simple Polynomial Puzzle for a solution.

This is a clever puzzle from the 1986 AIME problems.

“The pages of a book are numbered 1 through *n*. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice?”

See the Additional Page Problem for the solution.

This is a slightly different mystery number puzzle from the December 2023 MathsJam Shout. It provides a simpler puzzle as a respite from the more challenging problems.

This is a slightly challenging problem from Dan Griller.

“Every pupil at the Euler Academy studies French or Spanish. At the start of the year, one third of the French students also studied Spanish, and 2 fifths of the Spanish students also studied French. After one term, six of the double-linguists dropped French, so that now only a quarter of the French students study Spanish. How many pupils are at the Euler Academy?”

Just to be clear, “French students” means Euler Academy pupils studying French, and similarly for “Spanish students.”

See the Language Students Puzzle for solution.

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?”

See Al the Chemist I for solution.

This is another long historical story from Sam Loyd with a puzzle attached.

“NOTICING THE HIGH price recently paid at auction for an autograph of General Grant reminds me to say that I am the proud possessor of what I believe to be the last signature made by General Grant.

The story connected with it introduces a somewhat pretty problem, and induces me to pay a tribute to Grant’s mathematical ability, at the expense of the many who have no love for figures. I take occasion here to say that while journeying through life and jostling up against all manner of people, the fact has been impressed upon my mind that with few exceptions all successful men were those who endowed with a ready faculty for correct mental arithmetic. On the other hand, there is a class of never-do-wells who guess or jump at conclusions in a reckless way, and cannot even figure up how much to pay on the dollar when the inevitable smash comes.

I could mention a dozen incidents connected with great men as illustrating their aptitude for correct calculations, but this one will suffice to call attention to Grant’s aptitude for figures.

We all remember the story of how he figured his way into West Point, after that memorable journey for a pound of butter, when he heard of the chance for a competitive examination. Professor Agnell, the master of mathematics at West Point, with whom I used to play chess, used to say that “Grant had a great love for mathematics and horses.”

Grant did love a horse and could pick out the good qualities at a glance, and, oh, my! how he despised a man who would abuse a dumb animal!

My story turns upon an incident as told by Ike Reed, of the old horse mart of Johnson & Reed, who gave me the autograph from their sales book of 1884, as photographed in the picture. During the last term of his Presidency General Grant returned from his afternoon drive and in a humorous but somewhat mortified way told Colonel Shadwick, who kept the Willard Hotel, that he had been passed on the road by a butcher cart in a way that made his crack team appear to be standing still. He said he would like to know who owned the horse and if it was for sale.

The horse was readily found and purchased from an unsophisticated German for half of what he would have asked had he known the purchaser was the President of the United States. The horse was of light color and was none other than Grant’s favorite horse, “Butcher Boy,” named after the incident mentioned. Well, some years later, after the Wall street catastrophe, which impaired the finances of the Grant family, Butcher Boy and his mate were sent to the auction rooms of Johnson & Reed, and sold for the sum of $493.68. Mr. Reed said he could have gotten twice as much for them if he had been permitted to mention their ownership. But General Grant positively prohibited the fact being made known. “Nevertheless,” said Reed. “you come out two per cent, ahead, for you make 12 per cent, on Butcher Boy and lose 10 per cent, on the other.”

“I suppose that is the way some people would figure it out.” replied the General, but the way he laughed showed that he was better at figures than some people, so I am going to ask our puzzlists to tell me what he got for each horse if he lost 10 per cent on one and made 12 per cent on the other, but cleared 2 per cent on the whole transaction?

It may be mentioned incidentally that General Grant stated that he had presented one of the horses to Mrs. Fred Grant, and as shown in the receipt signed for her.”

See the Butcher Boy Problem for solution.

This is an initially mind-boggling problem from the 1995 American Invitational Mathematics Exam (AIME).

“Find the last three digits of the product of the positive roots of

”

See Log Lunacy for solution.

This is another physics-based problem from Colin Hughes’s *Maths Challenge* website (mathschallenge.net) that may take a bit more thought.

“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s^{2} until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.

Assuming no air resistance and taking *g* = 9.8 m/s^{2}, how long does it take to reach its maximum height?”

I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.

See the Fireworks Rocket for solutions.

This math problem from Colin Hughes’s *Maths Challenge* website (mathschallenge.net) hearkens back to basic physics.

“A boy drops a stone down a well and hears the splash from the bottom after three seconds. Given that sound travels at a constant speed of 300 m/s and the acceleration of the stone due to gravity is 10 m/s^{2}, how deep is the well?”

See the Falling Sound Problem for solutions.

I have just finished reading a most remarkable book by Alec Wilkinson, called *A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age*. I had read an essay of his in the *New Yorker* that turned out to be essentially excerpts from the book. I was so impressed with his descriptions of mathematics and intrigued by the premise of a mature adult in his 60s revisiting the nightmare of his high school experience with mathematics that I was eager to see if the book was as good as the essay. It was, and more.

The book is difficult to categorize—it is not primarily a history of mathematics, as suggested by Amazon. But it is fascinating on several levels. There is the issue of a mature perspective revisiting a period of one’s youth; the challenges of teaching a novice mathematics, especially a novice who has a strong antagonism for the subject; and insights into why someone would want to learn a subject that can be of no “use” to them in life, especially their later years.

Wilkinson has a strong philosophical urge; he wanted to understand the role of mathematics in human knowledge and the perspective it brought to life. He was constantly asking the big questions: is mathematics discovered or invented, what is the balance between nature and nurture, why does mathematics seem to describe the world so well, what is the link between memorization and understanding, how do you come to understand anything?