This is a simple problem from Five Hundred Mathematical Challenges:
“Problem 24. Let P be the center of the square constructed on the hypotenuse AC of the right-angled triangle ABC. Prove that BP bisects angle ABC.”
See the Center of Square Problem
Category Archives: Puzzles and Problems
Old Hook Puzzle
Here is another, more challenging, problem from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).
“An event that occurred during The Adventure of the Wandering Bishops inspired Holmes to devise a particularly tricky little mental exercise for my ongoing improvement. There were times when I thoroughly appreciated and enjoyed his efforts, and times when I found them somewhat unwelcome. I’m afraid that this was one of the latter occasions. It had been a bad week.
‘Picture three farmers,’ Holmes told me. ‘Hooklanders. We’ll call them Ern, Ted, and Hob.’
‘If I must,’ I muttered.
‘It will help,’ Holmes replied. ‘Ern has a horse and cart, with an average speed of eight mph. Ted can walk just one mph, given his bad knee, and Hob is a little better at two mph, thanks to his back.’
‘A fine shower,’ I said. ‘Can’t I imagine them somewhat fitter?’
‘Together, these worthies want to go from Old Hook to Coreham, a journey of 40 miles. So Ern got Ted in his cart, drove him most of the way, and dropped him off to walk the rest. Then he went back to get Hob [who was still walking], and took him into Coreham, arriving exactly as Ted did. How long did the journey take?’
Can you find a solution?”
I added the statement in brackets. I initially thought Hob waited in Old Hook until Ted fetched him. But the solution indicated that was not the case. So I realized Hob had started out at the same time as the others. The solution has some hairy arithmetic. Even knowing the answer it is difficult to do the computations without a mistake.
See the Old Hook Puzzle for solutions.
Moving Up
Here is a simple problem from an old Futility Closet posting.
“My wife and I walk up an ascending escalator. I climb 20 steps and reach the top in 60 seconds. My wife climbs 16 steps and reaches the top in 72 seconds. If the escalator broke tomorrow, how many steps would we have to climb?”
See Moving Up for solutions.
Catching the Thief
This typical problem from the prolific H. E. Dudeney may be a bit tricky at first.
“104.—CATCHING THE THIEF.
“Now, constable,” said the defendant’s counsel in cross-examination,” you say that the prisoner was exactly twenty-seven steps ahead of you when you started to run after him?”
“Yes, sir.”
“And you swear that he takes eight steps to your five?”
“That is so.”
“Then I ask you, constable, as an intelligent man, to explain how you ever caught him, if that is the case?”
“Well, you see, I have got a longer stride. In fact, two of my steps are equal in length to five of the prisoner’s. If you work it out, you will find that the number of steps I required would bring me exactly to the spot where I captured him.”
Here the foreman of the jury asked for a few minutes to figure out the number of steps the constable must have taken. Can you also say how many steps the officer needed to catch the thief?”
See Catching the Thief for solutions.
Geometric Puzzle Magnificence
Here is yet another collection of beautiful geometric problems from Catriona Agg (née Shearer). For some reason I found these a bit more challenging than the previous ones. Some of them required more time to “see” the breakthrough.
See Geometric Puzzle Magnificence (revised)
(Update 11/13/2022) Problem #3 Solution Corrected Continue reading
Wine Into Water Problem
Here is a challenging problem from the 1874 The Analyst.
“A cask containing a gallons of wine stands on another containing a gallons of water; they are connected by a pipe through which, when open, the wine can escape into the lower cask at the rate of c gallons per minute, and through a pipe in the lower cask the mixture can escape at the same rate; also, water can be let in through a pipe on the top of the upper cask at a like rate. If all the pipes be opened at the same instant, how much wine will be in the lower cask at the end of t minutes, supposing the fluids to mingle perfectly?
— Communicated by Artemas Martin, Mathematical Editor of Schoolday Magazine, Erie, Pennsylvania.”
I found the problem in Benjamin Wardhaugh’s book where he describes The Analyst:
“Beginning in 1874 and continuing as Annals of Mathematics from 1884 onward, The Analyst appeared monthly, published in Des Moines, Iowa, and was intended as “a suitable medium of communication between a large class of investigators and students in science, comprising the various grades from the students in our high schools and colleges to the college professor.” It carried a range of mathematical articles, both pure and applied, and a regular series of mathematical problems of varying difficulty: on the whole they seem harder than those in The Ladies’ Diary and possibly easier than the Mathematical Challenges in the extract after the next. Those given here appeared in the very first issue.”
I tailored my solution after the “Diluted Wine Puzzle”, though this problem was more complicated. Moreover, the final solution must pass from discreet steps to continuous ones.
There is a bonus problem in a later issue:
“19. Referring to Question 4, (No. 1): At what time will the lower cask contain the greatest quantity of wine?
—Communicated by Prof. Geo. R. Perkins.”
See the Wine Into Water Problem for solutions.
Equitable Slice Problem
This is another Brainteaser from the Quantum math magazine .
“How can a polygonal line BDEFG be drawn in a triangle ABC so that the five triangles obtained have the same area?”
I found this problem rather challenging, especially when I first tried to solve it analytically (using hyperbolas). Eventually I arrived at a procedure that would accomplish the result. (revised)
See the Equitable Slice Problem (revised)
(Update 9/22/2021) I goofed. I erroneously and foolishly thought Quantum had not solved the problem. Upon a closer reading I see what they were getting at and revised the posting.
Nested Polygons Puzzle
These two interesting problems were posed on MEI’s MathsMonday site on 3 February 2020 and 2 March 2020, respectively. MEI and readers posted various approaches, but I used a method suggested by another problem whose origin I no longer recall.
See the Nested Polygons Puzzle for solutions.
Three Runners Puzzle
Here is another problem from Presh Talwalkar which he says is adapted from India’s Civil Services Exam.
“There are three runners X, Y, and Z. Each runs with a different uniform speed in a 1000 meters race. If X gives Y a start of 50 meters, they will finish the race at the same time. If X gives Z a start of 69 meters, they will finish the race at the same time. Suppose Y and Z are in a [1000 meter] race. How much of a start should Y give to Z so they would finish the race at the same time?”
Even though Talwalkar’s original graphic showed all the runners in a 1000 meter race, it was not immediately clear to me from the wording that the race between Y and Z was also 1000 meters. But that was the case, so I made it explicit.
See the Three Runners Puzzle for solutions.
Rock, Paper, Scissors Problem
Here is another problem from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).
“Wiggins grinned at me. ‘You’ve not played Rock Paper Scissors before, Doctor?’
‘Doesn’t ring a bell,’ I told him.
‘Two of you randomly pick one of the three, and shout your choice simultaneously. There are hand gestures, too. If you both get the same, it’s a draw. Otherwise, scissors beats paper, paper beats rock, and rock beats scissors.’
‘So it’s a way of settling an argument,’ I suggested.
‘You were brought up wrong, Doctor,’ Wiggins said gravely. ‘Look, try it this way. I played a series of ten games with Alice earlier. I picked scissors six times, rock three times, and paper once. She picked scissors four times, rock twice, and paper four times. None of our games were drawn.’ He glanced at Holmes, who nodded. ‘So then, Doctor. What was the overall score for the series?’ ”
See the Rock Paper Scissors Problem for solutions.
(Update 7/29/2021) This problem in a different guise was presented by Futility Closet (7/28/2021) and attributed to Yoshinao Katagiri in Nobuyuki Yoshigahara’s Puzzles 101: A Puzzlemaster’s Challenge, 2004.