Category Archives: Puzzles and Problems

Lunchtime at the Fish Pond

This is a problem from the 629 AD work of Bhaskara I, a contemporary of Brahmagupta.

“A fish is resting at the northeast corner of a rectangular pool. A heron standing at the northwest corner spies the fish. When the fish sees the heron looking at him he quickly swims towards the south (in a southwesterly direction rather than due south). When he reaches the south side of the pool, he has the unwelcome surprise of meeting the heron who has calmly walked due south along the side and turned at the southwest corner of the pool and proceeded due east, to arrive simultaneously with the fish on the south side. Given that the pool measures 12 units by 6 units, and that the heron walks as quickly as the fish swims, find the distance the fish swam.”

Answer.

See Lunchtime at the Fish Pond for a solution.

Stacked Rhombuses Puzzle

This is a puzzle from Talwalkar’s set of “Impossible Puzzles with Surprising Solutions.”

“Call this puzzle the leaning tower of rhombi.

There are 5 isosceles triangles, aligned along their bases, with base lengths of 12, 13, 14, 15, 16 cm. The 10 quadrilaterals above are in rows of 4, 3, 2, and 1. Each quadrilateral is a rhombus, and the top of the tower is a square. What is the area of the square?”

Answer.

See Stacked Rhombuses Puzzle for solutions.

Evaporating Pool Problem

This is a fairly straight-forward problem from A+ Click.

“The water from an open swimming pool evaporates at a rate of 5 gallons per hour in the shade and 15 gallons per hour in the sun.  If the pool loses 8,400 gallons in June and there were no clouds, what is the average duration of night during that month?”

Answer Choices:     6 hours     8 hours     10 hours     12 hours

Answer.

See Evaporating Pool Problem for solutions.

Two Men Meet

This is another problem from the c.100AD Chinese mathematical work, Jiǔ zhāng suàn shù (The Nine Chapters on the Mathematical Art) found at the MAA Convergence website Convergence.

“A square walled city measures 10 li on each side.  At the center of each side is a gate.  Two persons start walking from the center of the city.  One walks out the south gate, the other the east gate.  The person walking south proceeds an unknown number of pu then turns northeast and continues past the corner of the city until they meet the eastward traveler.  The ratio of the speeds for the southward and eastward travelers is 5:3.  How many pu did each walk before they met? [1 li = 300 pu]”

Answer.

See Two Men Meet for a solution.

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.

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