Author Archives: Jim Stevenson

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.

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.

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.

A Voice From the Past

… I know I feel differently in the morning, after seeing the pictures and reading the stories of lives arbitrarily snuffed out and people who had little reduced to people who have nothing at all. But tonight I’m going to bed in a vicious mood, with a stomach so full of contempt for this poisoned republic and its brain-dead citizens that I can taste it in my mouth, like bile.

And the only thing I can think to ask God—if she does exists—is why, just for once, can’t you smite the wicked instead of the innocent?

Posted by billmon at 01:09 AM

(Billmon’s full post can now be found on the Internet Wayback Machine at  https://web.archive.org/web/20051001001007/http://billmon.org/)

This 24 September 2005 post was made during the height (or depths) of the Iraq War during the Bush-Cheney regime.  Also recall the other Bush debacle, Hurricane Katrina, was just the month before, August 23–31, 2005.  We thought we were in the depths of depravity and had no inkling that we were only at the higher rings of hell.

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