This is another puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.
“A mysterious square has materialized in the middle of the MCG, hovering in mid-air. The heights above the ground of three of its corners are 13, 21 and 34 metres. The fourth corner is higher still. How high?”
See the Floating Square Puzzle for solutions.
(Update 8/13/2023) Alternative Solution
Oscar Rojas has an alternative solution:
This problem can be solved by simple arithmetic: Assume that the square is lowered 13 m, then first corner is at ground level, second at 8 m and third at 21 m. The center of the square is at average altitude of opposite corners, that is 29/2 m, then the fourth corner is at 29 m. Now bring the square to initial altitude and the fourth corner is at 42m.
This collection of alternative solutions has reminded me of another great example of a variety of ways of solving a problem I recently came across in one of James Tanton’s Youtube videos. Its called A Lovely Geometry Puzzle and offers at least 5 ways to solve this puzzle, which is to find the fraction of the whole square taken up by the sum of the areas of the infinite sequence of blue shaded triangles.
Does the order of the corners affect the answer. Now the two highest corners are positioned diametrically.
I enjoy your Puzzles.
No the order should not matter. The lowest and highest will still have that property, and the two middle ones could be in either order, clockwise or counterclockwise. I am not sure what “diametrically” means in the case of a square vs. a circle. If you mean diagonally, the only way they can be higher than the other two is if they are at equal height and the square is horizontal to the ground.