Category Archives: Puzzles and Problems

Cat and Mouse

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This is a classic puzzle from Boris Kordemsky’s 1972 Moscow Puzzles.

“Purrer has decided to take a nap.  He dreams he is encircled by 13 mice: 12 gray and 1 white.  He hears his owner saying: “Purrer, you are to eat each thirteenth mouse, keeping the same direction.  The last mouse you eat must be the white one.”  Which mouse should he start from [eat first]?”

Answer.

See Cat and Mouse for a solution.

The Josephus Problem

This famous Josephus Problem presented on Youtube is somewhat different from the Cat and Mice puzzle, but still has similarities.  An article by Jay Bennett discussing the problem was published in Popular Mechanics in 2016.

 

Penn and Teller – Spelling Cards

It turns out that Penn and Teller have performed another magic trick recently that is based on mathematical principles and so is more or less self-working.  It is a more complicated version of the Cat and Mice puzzle, which I have dubbed the “Spelling Cards” trick. Continue reading →

Fill in the Blanks

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This is a fun puzzle from John Bassey at Puzzle Sphere.

“The diagram shows a heptagon with three circles on each side. Some circles already have the numbers 8 to 14 filled in, while the remaining circles need to be filled with the numbers 1 to 7. Each circle must contain one number, and the sum of the numbers in every set of three circles along a line must be the same.  Arrange the numbers!!!”

Answer.

See Fill in the Blanks for a solution.

Chinese Quadrilateral Puzzle

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This is another intimidating puzzle from Presh Talwalkar:

“Thanks to Eric from Miami for suggesting this problem and sending a solution!

From a 5th grade Chinese textbook: In the quadrilateral ABCD, angle A = 90°, angle ABD = 40°, angle BDC = 5°, angle C = 45°, and the length of AB is 6. Find the area of the quadrilateral ABCD.”

Answer.

See the Chinese Quadrilateral Puzzle for solutions.

Ubiquitous 60 Degree Problem

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This is an interesting problem from the Canadian Mathematical Society’s 2001 Olymon.

“Suppose that XTY is a straight line and that TU and TV are two rays emanating from T for which XTU = UTV = VTY = 60º. Suppose that P, Q and R are respective points on the rays TY, TU and TV for which PQ = PR. Prove that QPR = 60º.”

See the Ubiquitous 60 Degree Problem

Tire Wear

Thanks to Futility Closet I discovered a new source of math puzzles: A+Click.

“A+ Click helps students become problem solvers. Free, without ads, no calculators, and without signing-up. The website features a graduated set of 16,000+ challenging problems for students in grades one through twelve, starting from the very simple to the extremely difficult. … The questions concentrate on understanding, spatial reasoning, usefulness, and problem solving rather than math rules and theorems. The problems include a short description and an illustration to help problem solvers visualize the model. The problems can be solved within one minute and without using a calculator.”

My only quibble with “The questions concentrate on understanding, spatial reasoning, usefulness, and problem solving rather than math rules and theorems.” is that by keeping explicit math notation  and concepts to a minimum, the use of symbolic algebra and calculus is muted and there is a whiff of the medieval reliance on mental verbal agility rather than the power of the new mathematics. 

Still the problems are imaginative and challenging.  Here is a good example.

“The rear tires of my car wear out after 40,000 miles, while the front tires are done after 20,000 miles.  Estimate how many miles I should drive before the tires (front and rear) are rotated to drive the maximal distance.

Answer Choices:    15,000 miles     12,000 miles     13,333 miles     16,667 miles”

(I admit solving these under a minute is a challenge, at which I often failed.  Ignoring time constraints allows for greater care and a more thorough mulling over the intricacies of the problem.  Yes, those who have mastered math can solve problems faster than those who have not, but real mastery of math requires an inordinate attention to details, and that requires time.)

Answer.

See Tire Wear for solutions.

NSA Track and Field Puzzle

This is a puzzle from Futility Closet.

“A puzzle by Steven T., a systems engineer at the National Security Agency, from the NSA’s September 2016 Puzzle Periodical:

Three athletes (and only three athletes) participate in a series of track and field events. Points are awarded for 1st, 2nd, and 3rd place in each event (the same points for each event, i.e. 1st always gets “x” points, 2nd always gets “y” points, 3rd always gets “z” points), with x > y > z > 0, and all point values being integers.

The athletes are named Adam, Bob, and Charlie.

  • Adam finished first overall with 22 points.
  • Bob won the Javelin event and finished with 9 points overall.
  • Charlie also finished with 9 points overall.

Question: Who finished second in the 100-meter dash (and why)?”

I thought this puzzle impossible at first.  There didn’t seem to be enough information to solve it.  But a bit of trial-and-error opened a path.

Answer.

See NSA Track and Field Puzzle for solutions.

Sphere and Plane Puzzle

This is another puzzle from BL’s Weekly Math Games.

“a + b + c = 2, and

a2 + b2 + c2 = 12

where a, b, and c are real numbers.  What is the difference between the maximum and minimum possible values of c?”

The original problem statement mentioned a fourth real number d, but I considered it a typo, since it was not involved in the problem.

Answer.

See Sphere and Plane Puzzle for a solution.

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 your intuition agree?)”

Answer

See Air Travel for a solution.

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