# Alcan Highway Problem

This work problem from Geoffrey Mott-Smith is a little bit tricky.

“An engineer working on the Alcan Highway was heard to say, “At the time I said I could finish this section in a week, I expected to get two more bulldozers for the job. If they had left me what machines I had, I’d have been only a day behind schedule. As it is, they’ve taken away all my machines but one, and I’ll be weeks behind schedule!”

How many weeks?”

See the Alcan Highway Problem for a solution.

# Shared Spaces Puzzle

This is a nice puzzle from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008.  It is more a logic puzzle than a geometric one.

“In the diagram, each question mark represents one of six consecutive whole numbers. The sum of the numbers in the triangle is 39, the sum of those in the square is 46 and the sum of those in the circle is 85.  What are the six numbers?”

See the Shared Spaces Puzzle

# A Nice Factorial Sum

This is another infinite series from Presh Talwalkar, but with a twist.

“This problem is adapted from one given in an annual national math competition exam in France. Evaluate the infinite series:

1/2! + 2/3! + 3/4! + …”

The twist is that Talwalkar provides three solutions, illustrating three different techniques that I in fact have used before in series and sequence problems.  But this time I actually found a simpler solution that avoids all these.  You also need to remember what a factorial is: n! =n(n – 1)(n – 2)…3·2·1.

See a Nice Factorial Sum for solutions.

# Sizing Up

This is another fairly simple puzzle from Futility Closet from a while ago (2014).

“Two lines divide this equilateral triangle into four sections. The shaded sections have the same area. What is the measure of the obtuse angle between the lines?”

See Sizing Up for solutions.

# Parallel Stroll Problem

This is a slightly challenging problem from the 1993 American Invitational Mathematics Exam (AIME).

“Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Find t, the amount of time in seconds, before Jenny and Kenny can see each other again.”

See the Parallel Stroll Problem for solutions.

# Date Night

This is a fairly straight-forward logic puzzle from Alex Bellos’s Monday Puzzle in The Guardian.

“When it comes to the world of mathematical puzzles, Hungary is a superpower. Not just because of the Rubik’s cube, the iconic toy invented by Ernő Rubik in 1974, but also because of its long history of maths outreach.

In 1894, Hungary staged the world’s first maths competition for teenagers, four decades before one was held anywhere else. 1894 also saw the launch of KöMaL, a Hungarian maths journal for secondary school pupils full of problems and tips on how to solve them. Both the competition and the journal have been running continuously since then, with only brief hiatuses during the two world wars.

This emphasis on developing young talent means that Hungarians are always coming up with puzzles designed to stimulate a love of mathematics. (It also explains why Hungary arguably produces, per capita, more top mathematicians than any other country.)

I asked Béla Bajnok, a Hungarian who is now director of American Mathematics Competitions, a series of competitions involving 300,000 students in the US, whether he knew of any puzzles that originated in Hungary. The first thing he said that came to mind was the ‘3-D logic puzzle’, a type of logic puzzle in which you work out the solution in a three dimensional box, rather than (as is the case with the standard version) in a two-dimensional grid. He said he had never seen this type of puzzle outside Hungary.

Below are two examples he created. You could solve these using an extended two dimensional grid. It’s more in the spirit of the question, however, to draw a three-dimensional one, like you are looking at three sides of a Rubik’s Cube.

Date night

Andy, Bill, Chris, and Daniel are out tonight with their dates, Emily, Fran, Gina, and Huong. We have the following information.

1. Andy will go to the opera
2. Bill will spend the evening with Emily,
3. Chris would not want to go out with Gina,
4. Fran will see a movie
5. Gina will attend a workshop.

We also know that one couple will see an art exhibit. Who will go out with whom, and what will they do?

See Date Night

# Meeting on the Bridge

Here is another Brainteaser from the Quantum math magazine.

“Nick left Nicktown at 10:18 A.M. and arrived at Georgetown at 1:30 P.M., walking at a constant speed. On the same day, George left Georgetown at 9:00 A.M. and arrived at Nicktown at 11:40 A.M., walking at a constant speed along the same road. The road crosses a wide river. Nick and George arrived at the bridge simultaneously, each from his side of the river. Nick left the bridge 1 minute later than George. When did they arrive at the bridge?”

See Meeting on the Bridge for solutions.

# Log Jam

Here is a tricky little logarithm problem from the 2021 Math Calendar.

“Find x, where

log2(log4(x)) = log4(log2(x))”

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

See Log Jam for a solution.

# Clock Connections Puzzle

This is an imaginative puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings for 2012.

“You draw a line connecting the 5 and 9 on a clock face, and another line connecting the 3 and 8. What is the angle between the two lines?”

See the Clock Connections Puzzle for solutions.

# Turning Wheels Puzzle

This is a thoughtful little problem from Posamentier’s and Lehmann’s Mathematical Curiosities.

“We have nine wheels touching each other with diameters successively increasing by 1 cm. Beginning with 1 cm as the smallest circle, and 9 cm for the largest circle, how many degrees does the largest circle turn when the smallest circle turns by 90°?”

See the Turning Wheels Puzzle for solutions.