Tag Archives: plane geometry

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.

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.

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.

Chinese Quadrilateral Puzzle

This is another intimidating puzzle from Presh Talwalkar:

“Thanks to Eric from Miami for suggesting this problem and sending a solution!

From a 5th grade Chinese textbook: In the quadrilateral ABCD, angle A = 90°, angle ABD = 40°, angle BDC = 5°, angle C = 45°, and the length of AB is 6. Find the area of the quadrilateral ABCD.”

Answer.

See the Chinese Quadrilateral Puzzle for solutions.