This is an A-level type problem by Dale Chapman found on Bluesky via Catriona Agg.
“Prove, using an algebraic method, that
(n + 1)3 – n3
is odd for all natural numbers n = 1, 2, 3, … .”
See A-Level Test Problem for a Solution
This is a Harvard University geometry puzzle from BL’s Math Games.
“These are two circles of radii 3 and 5. The black line of length 10 connects the centres of the two circles. The blue line is a common tangent to both circles. The question is: what’s the length of the blue line?”
See Tangent to Circles Problem for a solution.
This is a problem from Presh Talwalkar.
“A bottle with a cylindrical base is partially filled with water and placed on a flat surface. The bottle’s height is 21 cm and the water’s height is 12 cm. The water is completely contained within the cylindrical body of the bottle. If the bottle is flipped upside down, then the water’s height is 15 cm. What fraction of the bottle is filled with water?”
See Water Bottle Puzzle for solutions.
Here is another problem from the “Brainteasers” section of the Quantum magazine.
“What is the angle between the hands of a clock at 7:38?”
See Degrees of Separation for solutions.
This is a problem from Federico Kereki in Puzzle Sphere.
“In this puzzle, we start with a quarter circle (from a circle with center P) and a semicircle. The PQ tangent from P to the semicircle is 9 units long. What is the area of the orange rectangle?”
See Two Arcs Puzzle for solutions.
This is a Tanya Khovanova problem via Alex Bellos.
“Tanya Khovanova is a luminary of the recreational mathematics scene. Tanya has now written her first book, Mathematical Puzzles and Curiosities, in collaboration with two other puzzle enthusiasts, Ivo David and Yogev Shpilman. It’s packed with fantastic new puzzles and twists on old ones.
Battleships. You are an admiral in the Navy, in charge of an important mission. You have two choices.
Which is the better option?”
See Battleship Problem for solutions.
This is an amazing piece of music posted on Futility Closet.
“Paul Wetzger’s 1900 composition “Avant et Retour” is a table canon — it’s designed to be placed on a table between two players so that one reads the score right side up while the other reads it upside down. The two halves produce one contrapuntal composition.”
This is not a musical palindrome, since playing it backwards is not the same as playing it forwards. But it seems to me to be more complex, since the reverse (upside down) playing has to tonally and rhythmically harmonize with the forward playing.
Futility Closet provided a synthesizer version of the piece. But I found a clarinet duet that I prefer, not just because I played a clarinet but because the clarinet version articulates slurs and staccatos that more closely follows the score.
See Forward and Back for a PDF.
(Update 9/6/2026) Both Scores
I realized it might be helpful to make a copy of the rotated version of the score so that both parts could be followed more easily. Here is the result.
Yet another puzzle from Presh Talwalkar.
“I thank Taka for the suggestion! He created the puzzle and posted it to Twitter.
In rectangle ABCD, side AB = 8. There are two circles: one tangent to AB and BC (and has point E), and another tangent to AD and CD (and has point F). The two circles are also tangent to each other. The line EF is parallel to AB, passes through the tangent point, and EF = 6. What is the length of BC?”
See the Boxed Circles Problem for a solution.
This is a problem from BL’s Math Games that was on a 2012 Oxford Admission Test.
“If two chords QP and RP on a circle of radius 1 meet in an angle θ at P, then the largest possible area of the shaded region RPQ is
(a) θ(1 + cos θ/2)__(b) θ + sin θ__(c) (π/2)( 1 – cos θ/2)__(d) θ”
See the Oxford Max Area Question for a solution.
This is another 
