This is a cute little problem I came across via James Tanton (spoiler alert) on Twitter by Ayush DM:

“Here is an old Watsapp problem. How high is the table? Also find the height of the cat and tortoise.”

This is a cute little problem I came across via James Tanton (spoiler alert) on Twitter by Ayush DM:

“Here is an old Watsapp problem. How high is the table? Also find the height of the cat and tortoise.”

This is a nice variation on the typical clock problem posed by Cary Mallon and retweeted by Henk Reuling:

“This clock has fallen on the floor, and unfortunately, there is no indication which way ‘up’ the clock should hang. However, both hands are pointing *precisely* at the [adjacent] minute marks. You can now work out what the time is.”

See the Fallen Clock Puzzle.

Here is another good problem from *Five Hundred Mathematical Challenges*:

**“Problem 100.** A hexagon inscribed in a circle has three consecutive sides of length *a* and three consecutive sides of length *b*. Determine the radius of the circle.”

This problem made me think of the Putnam Octagon Problem. Again my approach might be considered a bit pedestrian. 500 Math Challenges had a slightly slicker solution.

See the Lop-sided Hexagon Problem

This is another challenging puzzle from Presh Talwalkar that seems difficult to know where to start.

“Given the figure shown at left, what is the value of x?”

See the Chord Progression Puzzle

Here is another Brain Bogglers problem from 1987 by Michael Stueben.

“A quadrilateral with sides three, two, and four units in length is inscribed in a circle of diameter five. What’s the length of the fourth side of the quadrilateral?”

Like a number of other Brain Bogglers this problem also uses an insight that makes the solution easy.

See the Quad in Circle Problem.

Here is another Brainteaser from the *Quantum* magazine.

“Prove that the area of the red portion of the star is exactly half the area of the whole star. (N. Avilov)”

This is a relatively simple problem, but I wanted to include it because of its cartoon. Its implied gentle post-Soviet humor reminded me of that strange decade in US-Russian affairs between the end of the Cold War and the rise of Putin in the 21st century. The strangeness was brought home when we had our annual security checks of our classified document storage. Being mostly anti-submarine warfare (ASW) material the main concern was that it would not fall into the hands of the Soviets. But with the “demise” of the Soviet Union in 1989 no one cared any more about the classification. After decades of painfully securing these documents we could not suddenly turn them loose and throw them into the public trash. So we kept them secure anyway. You can imagine how we old cold-warriors feel about the current regime.

That is not to say that I didn’t welcome the thaw. Russian literature, both classical and even “Soviet realism”, as well as Russian cinema, is some of the world’s best. And Russian mathematicians have always been superior, and especially adept at communicating with novices. The collaboration of the American mathematicians and *Kvant* contributors in *Quantum* produced excellent results during the thaw. It is unfortunate that it could not survive the rise of Putin and his oligarchs.

See the Red Star

In looking through some old files I came across a math magazine I had bought in 1998. It was called *Quantum* and was published by the National Science Teachers Association in collaboration with the Russian magazine *Kvant* during the period 1990 to 2001 (coinciding with the Russian thaw, which in the following age of Putin seems eons ago). Fortunately, they are all online now. Besides some fascinating math articles the magazine contains a column of “Brainteasers.” Here is one of them:

“Alice used to walk to school every morning, and it took 20 minutes for her from door to door. Once on her way she remembered she was going to show the latest issue of *Quantum* to her classmates but had forgotten it at home. She knew that if she continued walking to school at the same speed, she’d be there 8 minutes before the bell, and if she went back home for the magazine she’d arrive at school 10 minutes late. What fraction of the way to school had she walked at that moment in time? (S. Dvorianinov)”

This is fairly straight-forward, but other problems in the magazine are a bit more challenging.

Here is another UKMT Senior Challenge problem from 2017, which has a straight-forward solution:

“The diagram shows a circle of radius 1 touching three sides of a 2 x 4 rectangle. A diagonal of the rectangle intersects the circle at P and Q, as shown.

What is the length of the chord PQ?

__A_√5____B_4/√5____C_√5 – 2/√5____D_5√5/6____E_2”

See the Circle in Slot Problem

A fun, relatively new, Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos) has puzzles couched in terms of the Holmes-Watson banter. The following problem is a variation on the Sam Loyd Tandem Bicycle Puzzle.

“ ‘Here’s something mostly unrelated for you to chew over, my dear Watson. Say you and I have a single bicycle between us, and no other transport options save walking. We want to get the both of us to a location eighteen miles distant as swiftly as possible. If my walking speed is five miles per hour compared to your four, but for some reason—perhaps a bad ligament—my cycling speed is eight miles per hour compared to your ten. How would you get us simultaneously to our destination with maximum rapidity?’

‘A cab,’ I suggested.

‘Without cheating,’ Holmes replied, and went back to tossing his toast in the air.”

See the Bicycle Problem

This is a stimulating little problem from the ever-creative James Tanton:

“An ant is at the east end of an infinite stretchy band, initially 2 ft long. Each day: ant walks 1 ft west on the band. Overnight while sleeping, band stretches to double its length (carrying ant westward as does so). Same routine each day/night. Will ant cover 99% of band’s length?”

(Ant from clipart-library.com)

See the Rubber Band Ant