The craziness of manipulating radicals strikes again. This 2006 four-star problem from Colin Hughes at Maths Challenge is really astonishing, though it takes the right key to unlock it.
“Problem Consider the following sequence:
For which values of [positive integer] n is S(n) rational?”
See Amazing Radical Sum for a solution.
Again we have a puzzle from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).
“Our pursuit of the dubious Alan Grey, whom we encountered during The Adventure of the Third Carriage, led Holmes and myself to a circular running track where, as the sun fell, we witnessed a race using bicycles. There was some sort of substantial wager involved in the matter, as I recall, and the track had been closed off specially for the occasion. This was insufficient to prevent our ingress, obviously.
One of the competitors was wearing red, and the other blue. We never did discover their names. As the race started, red immediately pulled ahead. A few moments later, Holmes observed that if they maintained their pace, red would complete a lap in four minutes, whilst blue would complete one in seven.
Having made that pronouncement, he turned to me. ‘How long would it be before red passed blue if they kept those rates up, old chap?’
Whilst I wrestled with the answer, Holmes went back to watching the proceedings. Can you find the solution?”
See the Track Problem for a solution.
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.”
See Table Tabby Tortoise Problem for solutions.
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 for a solution.
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 for solutions.
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 for solutions.
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 for a solution.
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.
See Calculating on the Way for solutions.
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 for a solution.
