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.
Degrees of Separation
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Odd Sum
This is another 
Two Arcs Puzzle
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.
Battleship Problem
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.
- To send a single ship whose chance of success is P per cent.
- To send two ships, each of whose chance of success is P/2 per cent. At least one ship needs to be successful for the mission to be a success.
Which is the better option?”
See Battleship Problem for solutions.
Forward and Back
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.
Boxed Circles Problem
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.
Oxford Max Area Question
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.
Challenging GRE Sum
This is an infinite series from the 2007 Graduate Record Exam (GRE) practice manual.
(A) e___(B) 2e___(C) (e + 1)(e – 1)___(D) e2___(E) ∞
See Challenging GRE Sum for a solution.
Conical Mountain Climb
This is a fun problem (#1455) from A+ Click.
“Gerry and Jane are at the foot of a conical-shaped mountain with a direct way to the peak of 8 km and the foot circle’s diameter of 8 km. They walk from point A to the diametrically opposite point B by the shortest route. Find the distance c from the foot of the mountain to the highest point of their route C.
Answer Choices:__2.34 km__2.66 km__3.34 km__3.66 km”
See Conical Mountain Climb for solutions.
Modular Arithmetic Puzzle
This is a rather remarkable problem from the 2025 Math Calendar.
“What is the value of
mod33(20252025)
where moda(b) is the remainder r < a after a divides b, that is,
b = ma + r
for some integer m.”
Recall that the answer is a day of the month.
See Modular Arithmetic Puzzle for a solution.