It’s time for another blindfold puzzle! As we’ve seen before, brain teasers like this one aren’t even proper puzzles when you can see what’s in front of you. But remove vision from the equation, and you’re left with a seriously devious riddle.

## Sunday Puzzle #29: Counting Coins

This week’s puzzle was recommended by reader Mikko P., who writes (with some minor edits and emphases added):

*You are blindfolded and brought to a table, on which rest fifty coins. Sixteen of them, you are told, are heads up. Thirty-four of them are tails up. Your task is to sort the coins into two groups, such that each group contains the same number of coins that are heads up. With your blindfold on, you cannot see if a coin is heads up. Neither can you feel a coin to determine its orientation. You can, however, move the coins and flip any number of them over.*

*How do you solve this puzzle when 10 coins are heads up? 27 coins?*

We’ll be back next week with the solution – and a new puzzle! Got a great brainteaser, original or otherwise, that you’d like to see featured? E-mail me with your recommendations. (Be sure to include “Sunday Puzzle” in the subject line.)

*Illustration by Jim Cooke*

## SOLUTION To Sunday Puzzle #28: String Around the Rod

Last week, I asked you to take a crack at a puzzle that once stumped 96% of some of the brightest high schoolers in the United States. The solution, I hinted strongly, was more easily determined than you might expect.

Indeed, what makes this puzzle so interesting—and, ultimately, satisfying—is that it requires no complex mathematical concepts to solve (this, despite appearing on a test designed to assess “advanced mathematics achievement of final-year students having taken advanced mathematics in 16 countries”). In fact, the crux of this puzzle has nothing to do with mathematics whatsoever; it has to do with how the problem is visualized.

By far the easiest way to solve this puzzle is to think of the rod not as a cylinder, but as a hollow tube that has been unraveled. Imagine a paper towel roll cut along its length and unfurled:

If this is your first time thinking of the problem in this way, stop reading immediately and go give the puzzle another shot. If you’re ready for the answer, keep scrolling.

If we ignore the glue stains and add some diagonal lines (these lines represent the string wrapped around the rod), it starts to become clear that this puzzle is, in fact, a simple geometry problem in disguise:

I’ll let reader John Milo Train explain the rest. Several of you provided correct solutions to last week’s puzzle (looking through the comments, it looks like Guywhothinksstuff and Harley-Beckett were the first), but Milo Train was the first person to submit a full, cogent solution (hooray, diagrams!):

When the rod is flattened into two dimensions, we find that the string can be thought of as the hypotenuse to four right triangles. Apply the Pythagorean theorem (a^{2} + b^{2} = c^{2}) to find the hypotenuse to one of the triangles, multiply by four, and—presto!—there’s your answer.

This is not the only way to solve the puzzle. As some commenters noted, it can also be solved with calculus. The authors of the Third International Mathematics and Science Report reference other solutions, and criteria for partial credit, on page 152 of their report (which you can download here):

Most of the students responding correctly used [the same approach as Milo Train], although a handful used variations (e.g., half of surface represented as a rectangle using eight congruent segments). Students receiving partial credit used the general approach, but made numerical errors in calculating the length of string. Students in all participating countries found this probelm very difficult. Only 10%, on average, provided a fully correct response, with another 2% , on average, receiving partial credit. Swedish students had the best performance, with 24% providing fully correct responses.

What’s great about this problem in particular—and this is something the authors of the report don’t mention—is that the triangles formed by the string aren’t just any right triangles. Each of them is a 3-4-5 triangle, one of a handful of “special” right triangles, so designated on account of there existing simple rules by which to solve their side-lengths and internal angles. The triangles in this puzzle are all Pythagorean triples, i.e. right triangles whose side lengths are all positive integers. The 3-4-5 triangle is probably the most well-known Pythagorean triple in the world; which means that, even if you were to blank on Pythagoream theorem, you could probably solve this problem pretty easily in your head—provided you knew to unroll the rod first. As the *New York Times*’ Edward Rothstein put it when he wrote about this problem back in 1998:

It is actually a beautiful solution, simple and startling enough so one almost has to take an object in hand and think about why it makes sense... Nothing could be a better illustration of the value of teaching a mathematical way of thinking. It requires different ways of examining objects; it might mean restating problems in other forms. It can demand a playful readiness to consider alternatives and enough insight to recognize patterns. Anyone who solves this problem starts to think differently about the world itself.

In his article, Rothstein ruminates nervously on whether America’s technological success is sustainable, in light of its educational mediocrity. Today, nearly two decades later, his concerns are as relevant as ever. But for the purposes of this column, and for its international audience, the most relevant bit of Rothstein’s piece is its headline, which tidily summarizes something I’ve tried to impress upon you all since the Sunday Puzzle column’s early days:

#### “It’s not just numbers or advanced science, it’s also knowing how to think.”

**Previous Weeks’ Puzzles**

- Can You Solve This Extremely Difficult
*Star Trek*Puzzle? - You Either Solve This Riddle, Or You Die
- Can You Solve ‘The Hardest Logic Puzzle In the World’?
- You’ll Need All 3 Clues To Solve This Puzzle
- Think You Know The Solution To This Classic Riddle? Think Again.
- 100 Lives Are On The Line In This Week’s Puzzle. How Many Can You Save?
- Can You Figure Out This Parking Lot’s Numbering System?
- To Solve This Riddle, Look To Your Family
- Solving This Puzzle Will Help You Grasp The True Nature Of Puzzles
- Can You Guess The Next Number In This Sequence?

*Contact the author at rtgonzalez@io9.com.*