Shy Angle Problem

Here is yet another problem from Presh Talwalkar. This one is rather elegant in its simplicity of statement and answer.

“Solve For The Angle – Viral Puzzle

I thank Barry and also Akshay Dhivare from India for suggesting this problem!  This puzzle is popular on social media. What is the measure of the angle denoted by a “?” in the following diagram? You have to solve it using elementary geometry (no trigonometry or other methods).  It’s harder than it looks.  I admit I did not solve it. Can you figure it out?”

See the Shy Angle Problem for solutions.

Loggers Problem

Here is another delightful problem from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).

“In Sussex, Holmes and I ran into a pair of woodcutters named Doug and Dave. There was an air of the unreliable about them—not helped by a clearly discernable aroma of scrumpy—but they nevertheless proved extremely helpful in guiding us to a particular hilltop clearing some distance outside of the town of Arundel. A shadowy group had been counterfeiting sorceries of a positively medieval kind, and all sorts of nastiness had ensued.

The Adventure of the Black Alchemist is not one that I would feel comfortable recounting, and if my life never drags me back to Chanctonbury Ring I shall be a happy man. But there is still some instructive material here. Whilst we were ascending our hill, Doug and Dave made conversation by telling us about their trade. According to these worthies, working together they were able to saw 600 cubic feet of wood into large logs over the course of a day, or split as much as 900 cubic feet of logs into chunks of firewood.

Holmes immediately suggested that they saw as much wood in the first part of the day as they would need in order to finish splitting it at the end of the day. It naturally fell to me to calculate precisely how much wood that would be.

Can you find the answer?”

See the Loggers Problem for solutions.

Barrier Minimal Path Problem

This is a nifty little problem from the Quantum math magazine.

“Two ants stand at opposite corners of a 1-meter square. A barrier was placed between them in the form of half a 1-meter square attached along the diagonal of the first square, as shown in the picture. One ant wants to walk to the other. How long is the shortest path?”

See the Barrier Minimal Path Problem for solutions.

Do I Avoid Kangaroos?

This is a fun logic puzzle from one of Ian Stewart’s many math collections.  I discovered that the problem actually is basically one of Lewis Carroll’s examples from an 1896 book:

1. The only animals in this house are cats.
2. Every animal that loves to gaze at the moon is suitable for a pet.
3. When I detest an animal, I avoid it.
4. No animals are meat-eaters, unless they prowl by night.
5. No cat fails to kill mice.
6. No Animals ever take to me, except those in this house.
7. Kangaroos are not suitable for pets.
8. Only meat-eaters kill mice.
9. I detest animals that do not take to me.
10. Animals that prowl at night love to gaze at the moon.

If all these statements are correct, do I avoid kangaroos, or not?

See Do I Avoid Kangaroos? for solutions.

Square Deal

Here is a simple Futility Closet problem from 2014.

“This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?”

See the Square Deal for solutions.

Curve Making Puzzle

Here is a problem from Five Hundred Mathematical Challenges that I indeed found quite challenging.

“Problem 235. Two fixed points A and B and a moving point M are taken on the circumference of a circle. On the extension of the line segment AM a point N is taken, outside the circle, so that lengths MN = MB. Find the locus of N.

Since one of the first hurdles I faced with this problem was trying to figure out what type of shape was being generated, I thought I would omit my usual drawings illustrating the problem statement.  There turned out to be a lot of cases to consider, but the result was most satisfying.  I also included the case when N is inside the circle.  Again Visio was my main tool to handle all the examples with the concomitant requirement to prove whatever Visio suggested.

See the Curve Making Puzzle

The Thoth Maneuver

This is a nice puzzle from Clifford Pickover in the 1996 Discover magazine’s Brain Bogglers.

“Thoth, ancient Egyptian god of wisdom and learning, has abducted Ahmes, a famous Egyptian scribe, in order to assess his intellectual prowess. Thoth places Ahmes before a large funnel set in the ground. It has a circular open­ing 1,000 feet in diameter, and its walls are quite slippery. If Ahmes attempts to enter the funnel, he will slip down the wall. At the bottom of the funnel is a sleep-inducing liquid that will instantly put Ahmes to sleep for eight hours if he touches it.

As shown in the illustration, there are two ankh-shaped towers. One stands on a cylindrical platform in the center of the fun­nel. The platform’s surface is at ground level. The distance from the platform’s surface to the liquid is 500 feet. The other ankh tower is on land, at the edge of the funnel.

Thoth hands Ahmes two objects: a rope 1,006.28 feet in length and the skull of a chicken. Thoth says to Ahmes, ‘If you are able to get to the central tower and touch it, we will live in harmony for the next millennium. If not, I will detain you for fur­ther testing. Please note that with each passing hour, I will decrease the rope’s length by a foot.’

How can Ahmes reach the central ankh tower and touch it? ”

See the Thoth Maneuver

Rufus Puzzle

Again we have a puzzle from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).  This one is quite a bit more challenging, at least for me.

“When Holmes and I met with Wiggins one afternoon, he was accompanied by a rather scrappy-looking mutt, who eyed me with evident suspicion.

‘This is Rufus,’ Wiggins said. ‘He’s a friend.’

‘Charmed,’ I said.

‘He’s very energetic,’ Wiggins told us. ‘Just this morning, he and I set out for a little walk.’

At the word ‘walk’, the dog barked happily.

‘When we set out, he immediately dashed off to the end of the road, then turned round and bounded back to me. He did this four times in total, in fact. After that, he settled down to match my speed, and we walked the remaining 81 feet to the end of the road at my pace. But it seems to me that if I tell you the distance from where we started to the end of the road, which is 625 feet, and that I was walking at four miles an hour, you ought to be able to work out how fast Rufus goes when he’s running.’

‘Indeed we should,’ said Holmes, and turned to look at me expectantly.

What’s the dog’s running speed?”

See the Rufus Puzzle for solutions.

Circle-Halving Zigzag Problem

This is another delightful Brainteaser from the Quantum math magazine.

“All the vertices of a polygonal line ABCDE lie on a circumference (see the figure), and the angles at the vertices B, C, and D are each 45°.

Prove that the area of the blue part of the circle is equal to the area of the yellow part. (V. Proizvolov)”

I especially liked this problem since I was able to find a solution different from the one given by Quantum.  Who knows how many other variations there might be.