Category Archives: Puzzles and Problems

Railway Crossing Problem

This is an interesting problem from the 1966 Eureka magazine.

“A railway and a road run together for seven miles from P to Q. Two miles from P there is a level crossing, which is closed one minute before, and opened one minute after, a train passes.

A train passes a Stationary car at P and travels on to Q at 60 m.p.h., and, forgetting to slow down, crashes at Q; the car passes the train as it crashes. Assuming that stopping for an instant from full speed loses the car one minute, of what speed must it be capable?”

Answer

See the Railway Crossing Problem for a solution.

100 Light Bulbs Puzzle

This is a classic puzzle from Presh Talwalkar.

“This puzzle has been asked as an interview question at tech companies like Google.

There are 100 lights numbered 1 to 100, all starting in the off position. There are also 100 people numbered 1 to 100. First, person 1 toggles every light switch (toggle means to change from off to on, or change from on to off). Then person 2 toggles every 2nd light switch, and so on, where person i toggles every ith light switch. The last person is person 100 who toggles every 100th switch.

After all 100 people have passed, which light bulbs will be turned on?”

I vaguely remembered the answer, which I confirmed after a few examples. But I didn’t remember an exact proof, so I thought I would give it a try.

Answer.

See 100 Light Bulbs Puzzle for solutions.

Sheep in Garden Problem

This is a slightly challenging problem from BL’s Math Games.

“In a square garden ABCD of side 10m, a sheep sets off from B and moves along BC at 30cm per minute.  At the same time, you set off from C and move along edge CD at 40cm per minute.  The question is, what’s the shortest distance between you and the sheep in meters?

This is somewhat an optimization problem because as you and the sheep move along the sides of the square at different rates, the distance in between varies as you can imagine.”

There’s at least one non-calculus solution and of course one calculus solution.

Answer.

See Sheep in Garden Problem for solutions.

Unlawful Distance

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

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

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