Tag Archives: algebra

Mystery Quadratic

Presh Talwalkar has an interesting new problem.

“Students and teachers found a recent test in New Zealand to be confusing and challenging for covering topics that were not taught in class.

For the equation below, find the value of k for which the equation has numerically equal but opposite signs (for example, 2 and –2):

The problem didn’t mention how old the students were, but the solution to another problem on the test indicates they needed to know calculus.

Answer.

See Mystery Quadratic for a solution.

Air Travel

This is a nice problem from Five Hundred Mathematical Challenges.

“Problem 62. A plane flies from A to B and back again with a constant engine speed.  Turn-around time may be neglected.  Will the travel time be more with a wind of constant speed blowing in the direction from A to B than in still air?  (Does you intuition agree?)”

Answer

See Air Travel for a solution.

Language Students Puzzle

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.”

Answer.

See the Language Students Puzzle for solution.

Al the Chemist I

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?”

Answer.

See Al the Chemist I for solution.

Butcher Boy Problem

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.”

Answer.

See the Butcher Boy Problem for solution.