This is another problem from BL’s Math Games.
“What fraction of the rectangle is colored? Assume that M and N are midpoints of the sides of the rectangle.”
That they are midpoints was not stated explicitly in the problem as given in front of the subscription wall, but from the comments it became evident this was the case.
Initially I actually assumed the line was positioned arbitrarily. What would be the solution in that case?
Answer to BL problem.
See ZigZag in Rectangle for a solution.
This problem is from BL’s Math Games.
“What’s the area of the red triangle?”
BL decided to see what kind of solution ChatGPT would come up with. After several tries and prompts it seemed to oblige. I don’t know what BL’s prompts were, and in the statement of the problem outside the subscription wall he never explicitly says what the problem is, namely, to find the area of the red triangle.
There also seems to be some ambiguity about the constraints on the problem, that is, how much of the appearance of the diagram should the solver assume?
See ChatGPT Problem for a solution.
This puzzle is from the Irishman Owen O’Shea.
“The following puzzle illustrates a beautiful mathematical relationship involving a rectangle of any size and a random point within that rectangle that most people, including mathematicians, are unaware of.
The figure shows a rectangular room. There is a matchbox located 6 feet from one corner of the room and 27 feet from the opposite corner. The matchbox is also located 21 feet from a third corner.
How far is the matchbox from the fourth corner?”
See the Spot in a Rectangle Problem for a solution.
This is a problem from the 1995 AIME problems.
“Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.”
See the Three Circles Problem for solutions.
This is another problem from Dan Griller.
“In the triangle ABC, CN and MB are straight lines, ÐCAB = 90° and CM = MA = AN = NB = 5. Find the exact area of the shaded region.”
See the Triangle Bow-tie Problem for a solution.
This is an earlier puzzle from Presh Talwalkar.
“A square pyramid has base PQRS and vertex O. Each edge has length equal to 20. Calculate the shortest distance along the outer surface of the pyramid from P to T, the midpoint of OR.”
See Shortest Pyramid Path for solutions.
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.”
See Lunchtime at the Fish Pond for a solution.
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?”
See Stacked Rhombuses Puzzle for solutions.
This is an old puzzle from Catriona Agg that I found on BL’s Math Games website.
“All four triangles are equilateral. What fraction of the largest triangle is shaded?”
This problem turned out to be both easy and unexpectedly challenging, at least for me.
See Four Equilateral Triangles for solutions.
