Category Archives: Puzzles and Problems

Unlawful Distance

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This is a puzzle from the A+Click site.

“There is a fault with the cruise control on Hank’s car such that the speed continuously and linearly increases with time.  When he starts off the speed is set to exactly 60 mph.  He is driving on a long straight route with the radio on at full blast and he is not paying any attention to his speed.  After 3 hours he notices that his speed has now reached 80 mph.  For how many miles did he drive above the state speed limit of 70 mph?

Answer Choices:            125 miles     112.5 miles     105 miles     99.5 miles”

Answer.

See Unlawful Distance for solutions.

Another Passing Train Puzzle

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This is another take on the passing train type puzzle from the Moscow Puzzles.

“A train moving 45 miles per hour meets and is passed by a train moving 36 miles per hour.  A passenger in the first train sees the second train take 6 seconds to pass him.  How long is the second train?”

Answer.

See Another Passing Train Puzzle for solutions.

Wittenbauer’s Parallelogram

This is a lovely result from Futility Closet.

“Draw an arbitrary quadrilateral and divide each of its sides into three equal parts. Draw a line through adjacent points of trisection on either side of each vertex and you’ll have a parallelogram.

Discovered by Austrian engineer Ferdinand Wittenbauer.”

Find a proof.

See Wittenbauer’s Parallelogram for a solution.

Two Squares in a Circle

This puzzle, from another set of seven challenges assembled by Presh Talwalkar, turned out to be very challenging for me.

“This is a fun problem I saw on Reddit AskMath. A circle contains two squares with sides of 4 and 2 cm that overlap at one point, as shown. What is the area of the circle?”

This took me quite a while to figure out, but I relied on another problem I had posted earlier.

Answer.

See Two Squares in a Circle for solutions.

Whose Bullet?

Here is a probability problem from BL’s Weekly Math Games.  Normally I am not a fan of such problems, but this one seemed fairly straight-forward for a change.

“I hit the target 75% of the time. You hit the target 25% of the time.  We aim at the same time, and only one bullet hits.  What’s the probability it came from me?

Now as for this puzzle, it would be tempting to think that I am 3 times as good at hitting the target, but I am not!”

Answer.

See Whose Bullet for a solution

Brick in Water Puzzle

I thought this puzzle, which was included among a set of seven challenges assembled by Presh Talwalkar, would be fairly straight-forward.

“A cube of 50 cm is filled halfway with water. A rectangular prism with a square base of 25 cm and a height around 50 cm is placed flat onto the base of the cube, as shown. By how much does the water level rise?

Thanks to Fahad Alomaim for the suggestion! This is translated from a Mawhiba curriculum question for 8th grade.”

But I got the wrong answer and found Talwalkar’s solution a bit hard to fathom at first.  Looks like I flunked 8th grade.

Answer.

See Brick in Water Puzzle for solutions.

The King of the Spiders

Continuing the logic thread, this is a nice logic problem from MathsJam Shout for April 2025.

“The king of the spiders has four servants, and the servants have either 6, 7, or 8 legs.  Servants with 7 legs always lie, and servants with 6 or 8 legs always tell the truth.

The king asks ‘How many legs do you four have in total?’, and the four spider servants (who are standing behind a table, so you can’t see their legs) answer 25, 26, 27, and 28, respectively.

Who is telling the truth?”

Answer.

See The King of the Spiders for a solution.

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