Battleship Problem

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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.

  1. To send a single ship whose chance of success is P per cent.
  2. 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?”

Answer.

See Battleship Problem for solutions.

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Forward and Back

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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.

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Boxed Circles Problem

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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?”

Answer.

See the Boxed Circles Problem for a solution.

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Oxford Max Area Question

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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”

Answer.

See the Oxford Max Area Question for a solution.

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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”

Answer.

See Conical Mountain Climb for solutions.

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Running in Circles

This is a tricky problem from the Scientific American.

“Miles and Walker begin a workout together at the starting line of a closed-loop track. They run in opposite directions, each at their own constant pace, and continue running until they are both at the starting line at the exact same time again. They pass each other 11 times before this happens (not counting the start or end). Walker is slower than Miles but still manages to complete more than one lap. How many laps does Walker complete?”

I got the wrong answer from just thinking about it, and it took a number of diagrams to see where I had made some false assumptions.

Answer.

See Running in Circles for solutions.

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The B List Puzzle

This is a puzzle from Futility Closet:

“A problem from the Eighth International Mathematical Olympiad, held in Sofia, Bulgaria, in July 1966 (contributed by the Soviet Union):

In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem A, the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. Of all students who solved just one problem, half did not solve problem A. How many students solved only problem B?”

Answer.

See The B List Puzzle for solutions.

Puzzles and Problems: Futility Closet, IMO, algebra

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