Category Archives: Puzzles and Problems

The Damaged Engine

Yet another train problem from H. E. Dudeney.

“We were going by train from Anglechester to Clinkerton, and an hour after starting an accident happened to the engine. We had to continue the journey at three-fifths of the former speed. It made us two hours late at Clinkerton, and the driver said that if only the accident had happened fifty miles farther on the train would have arrived forty minutes sooner. Can you tell from that statement just how far it is from Anglechester to Clinkerton?”

Answer.

See the Damaged Engine for a solution.

Triangle Acute-Angle Problem

Catriona Shearer retweeted the following problem from Antonio Rinaldi‏ @rinaldi6109

“My little contribution to @Cshearer41 October 7, 2018
A point D is randomly chosen inside the equilateral triangle ABC. Determine the probability that the triangle ABD is acute-angled.”

Answer.

See Triangle Acute-Angle Problem for a solution.

Parallelogram Cosine Problem

Another challenging problem from Presh Talwalkar. I certainly could not have solved it on a timed test at the age of 16.

“One Of The Hardest GCSE Test Questions – How To Solve The Cosine Problem

Construct a hexagon from two congruent parallelograms as shown. Given BP = BQ = 10, solve for the cosine of PBQ in terms of x.

This comes from the 2017 GCSE exam, and it confused many people. I received many requests to solve this problem, and I thank Tom, Ben, and James for suggesting it to me.”

Answer.

See the Parallelogram Cosine Problem for solutions.

Hitting the Target

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2012:

“Tom and Geri have a competition. Initially, each player has one attempt at hitting a target. If one player hits the target and the other does not then the successful player wins. If both players hit the target, or if both players miss the target, then each has another attempt, with the same rules applying. If the probability of Tom hitting the target is always 4/5 and the probability of Geri hitting the target is always 2/3, what is the probability that Tom wins the competition?

______A 4/15______B 8/15______C 2/3______D 4/5______E 13/15”

Answer.

See Hitting the Target for solutions.

Kissing Angles

I really was trying to stop including Catriona Shearer’s problems, since they are probably all well-known and popular by now. But this is another virtually one-step-solution problem that again seems impossible at first. Many of her problems entail more steps, but I am especially intrigued by the one-step problems.

“What’s the sum of the two marked angles?”

Answer.

See Kissing Angles for a solution.

Ladies’ Diary Problem

An amazing publication was conceived primarily for women at the beginning of the 18th century in 1704 and was called The Ladies’ Diary or Woman’s Almanack. What made it even more remarkable was that each issue contained mathematical problems whose solutions from the readers were provided in the next issue. One particularly sharp woman was Mary Wright (Mrs. Mary Nelson). This is one of her problems:

“VIII. Question 72 by Mrs. Mary Nelson
(proposed in 1719, answered in 1720)

A prize was divided by a captain among his crew in the following manner: the first took 1 pound and one hundredth part of the remainder; the second 2 pounds and one hundredth part of the remainder; the third 3 pounds and one hundredth part of the remainder; and they proceeded in this manner to the last, who took all that was left, and it was then found that the prize had by this means been equally divided amongst the crew. Now if the number of men of which the crew consisted be added to the number of pounds in each share, the square of that sum will be four times the number of pounds in the chest: How many men did the crew consist of, and what was each share?”

What makes this problem nice is that it does have a clean answer, contrary to most of the problems in The Ladies’ Diary.

Answer.

See the Ladies’ Diary Problem for solutions.

(Update 5/6/2019) Continue reading →

Two Trains – London to Liverpool

This is another train puzzle from H. E. Dudeney, which is fairly straight-forward.

“I put this little question to a stationmaster, and his correct answer was so prompt that I am convinced there is no necessity to seek talented railway officials in America or elsewhere. Two trains start at the same time, one from London to Liverpool, the other from Liverpool to London. If they arrive at their destinations one hour and four hours respectively after passing one another, how much faster is one train running than the other?”

Answer.

See Two Trains – London to Liverpool for a solution.

Challenging Sum

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2009:

“Four positive integers a, b, c, and d are such that

abcd + abc + bcd + cda + dab + ab + bc + cd + da + ac + bd + a + b + c + d = 2009.

What is the value of a + b + c + d?

A 73_________B 75_________C 77_________D 79_________E 81”

Answer.

See the Challenging Sum for a solution.

(Update 4/17/2019) Continue reading →

Two Block Incline Puzzle

Since everyone by now who has any interest has gone directly to Catriona Shearer’s Twitter account for geometric puzzles, I was not going to include any more. But this one with its one-step solution is too fine to ignore and belongs with the “5 Problem” as one of the most elegant.

“Two squares sit on the hypotenuse of a right-angled triangle. What’s the angle?”

Answer.

See the Two Block Incline Puzzle for a solution.

(Update 4/26/2019) Continue reading →

Magic Parallelogram

I came across this problem in Alfred Posamentier’s book, but I remember I had seen it a couple of places before and had never thought to solve it. At first, it seems like magic.

In any convex quadrilateral (line between any two points in the quadrilateral lies entirely inside the quadrilateral) inscribe a second convex quadrilateral with its vertices on the midpoints of the sides of the first quadrilateral. Show that the inscribed quadrilateral must be a parallelogram.

See the Magic Parallelogram.

(Update 5/15/2020) Continue reading →

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