Category Archives: Puzzles and Problems

Triangle Projection Problem

This is a Maths Item of the Month (MIOM) problem that seems opaque at first.  (“The Maths Item of the Month is a monthly problem aimed at teachers and students of GCSE and A level Mathematics.”)

“Two fixed circles, C1 and C2, intersect at A and B.  P is on C1.  PA and PB produced meet C2 at A’ and B’ respectively.  How does the length of the chord A’B’ change as P moves?”

Just start noticing relationships and the answer falls out nicely.

(MIOM problems often appear on MathsMonday and are also produced by Mathematics Education Innovation (MEI).)

Answer.

See the Triangle Projection Problem for a solution.

Broken Diagonal Problem

This is a nice problem from the UKMT Senior Mathematics Challenge for 2022:

“Five line segments of length 2, 2, 2, 1 and 3 connect two corners of a square as shown in the diagram. What is the shaded area?

A 8____B 9____C 10____D 11____E 12”

The pleasure of solving this problem may be lessened if one is under a time crunch, as is the case with all these timed tests.

Answer.

See the Broken Diagonal Problem for solutions.

Road Construction Problem

This is an interesting problem from the Scottish Mathematics Council (SMC) 2014 Senior  Math Challenge .

“Two straight sections of a road, each running from east to west, and located as shown, are to be joined smoothly by a new roadway consisting of arcs of two circles of equal radius. The existing roads are to be tangents at the joins and the arcs themselves are to have a common tangent where they meet.  Find the length of the radius of these arcs.”

Answer.

See the Road Construction Problem for solutions.

Falling Sound Problem

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/s2, how deep is the well?”

Answer.

See the Falling Sound Problem for solutions.

Neuberg’s Theorem

This turned out to be a challenging puzzle from the 1980 Canadian Math Society’s magazine, Crux Mathematicorum.

“Proposed by Leon Bankoff, Los Angeles, California.

Professor Euclide Paracelso Bombasto Umbugio has once again retired to his tour d’ivoire where he is now delving into the supersophisticated intricacies of the works of Grassmann, as elucidated by Forder’s Calculus of Extension. His goal is to prove Neuberg’s Theorem:

If D, E, F are the centers of squares described externally on the sides of a triangle ABC, then the midpoints of these sides are the centers of squares described internally on the sides of triangle DEF.  [The accompanying diagram shows only one internally described square.]

Help the dedicated professor emerge from his self-imposed confinement and enjoy the thrill of hyperventilation by showing how to solve his problem using only highschool, synthetic, Euclidean, ‘plain’ geometry.”

Alas, my plane geometry capability was inadequate to solve the puzzle that way, so I had to resort to the sledge hammer of analytic geometry, trigonometry, and complex variables.

See Neuberg’s Theorem

Refabulating Widgets

This is a work problem from Geoffrey Mott-Smith from 1954.

“ ‘If a man can do a job in one day, how long will it take two men to do the job?’

No book of puzzles, I take it, is complete without such a question. I will not blame the reader in the least if he hastily turns the page, for I, too, was annoyed by “If a man” conundrums in my schooldays. Besides, the answer in the back of the book was always wrong. Everybody knows it will take the two men two days to do the job, because they will talk about women and the weather, they will argue about how the job is to be done, they will negotiate as to which is to do it. In schoolbooks the masons and bricklayers are not men, they are robots.

Strictly on the understanding that I am really talking about robots, I will put it to you:

If a tinker and his helper can refabulate a widget in 2 days, and if the tinker working with the apprentice instead would take 3 days, while the helper and the apprentice would take 6 days to do the job, how long would it take each working alone to refabulate the widget?”

Answer.

See Refabulating Widgets for a solution.

Skating Rendezvous Problem

This is a fun problem from the 1989 American Invitational Mathematics Exam (AIME).

“Two skaters, Allie and Billie, are at points A and B, respectively, on a flat, frozen lake. The distance between A and B is 100 meters. Allie leaves A and skates at a speed of 8 meters per second on a straight line that makes a 60° angle with AB. At the same time Allie leaves A, Billie leaves B at a speed of 7 meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?”

Answer.

See the Skating Rendezvous Problem for solutions.

Mountain Climbing Puzzle

This puzzle from the Scottish Mathematical Council (SMC) Middle Mathematics Challenge has an interesting twist to it.

“Two young mountaineers were descending a mountain quickly at 6 miles per hour. They had left the hostel late in the day, had climbed to the top of the mountain and were returning by the same route.  One said to the other “It was three o’clock when we left the hostel. I am not sure if we will be back before nine o’clock.” His companion replied “Our pace on the level was 4 miles per hour and we climbed at 3 miles per hour. We will just make it.” What is the total distance they would cover from leaving the hostel to getting back there?”

Answer.

See the Mountain Climbing Puzzle for solutions.

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