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 →

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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.

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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.

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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

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Martin Luther King Day 2025

From Futility Closet:

A Lesson

John Alexander Smith, Waynflete Professor of Moral and Metaphysical Philosophy at Oxford, opened a course of lectures in 1914 with these words:

“Gentlemen — you are now about to embark upon a course of studies which will occupy you for two years. Together, they form a noble adventure. But I would like to remind you of an important point. Some of you, when you go down from the University, will go into the Church, or to the Bar, or to the House of Commons, to the Home Civil Service, to the Indian and Colonial Services, or into various professions. Some may go into the Army, some into industry and commerce; some may become country gentlemen. A few — I hope a very few — will become teachers or dons. Let me make this clear to you. Except for the last category, nothing that you will learn in the course of your studies will be of the slightest possible use to you in after life — save only this — that if you work hard and intelligently you should be able to detect when a man is talking rot, and that, in my view, is the main, if not the sole, purpose of education.”

See Martin Luther King Day for a PDF version.

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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.

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Double Areas Puzzles

A while ago James Tanton provided a series of puzzles:

Puzzle #1   At what value between 0 and 1 does a horizontal line at that height produce two regions of equal area as shown on the graph of y = x2?

Puzzle #2   A horizontal line is drawn between the lines y = 0 and y = 1, dividing the graph of y = x2 into two regions as shown. At what height should that line be drawn so that the sum of the areas of these two regions is minimal?

Puzzle #3   A horizontal line is drawn between the lines y = 0 and y = 1, dividing the graph of y = xn into two regions as shown (n > 0). At what height should that line be drawn so that the sum of the areas of these two regions is minimal? Does that height depend on the value of n?

Puzzle #4   What horizontal line drawn between y = 0 and y = 1 on the graph of y = 2√x – 1 minimizes the sum of the two shaded areas shown?

See Double Areas Puzzles for solutions.

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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.

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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.

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