Alcuin's triangular city problem is Problem 28 from Propositiones ad Acuendos Juvenes.

There is a triangular city which has one side of 100 feet, another side of 100 feet, and a third of 90 feet. Inside of this, I want to build a structure of houses, however, in such a way that each house is 20 feet in length, 10 feet in width. Let him say, he who can, How many houses should be contained [within this structure]?

I have learned about this problem from a B.Sc. thesis at a Slovak university. This thesis was didactics-oriented, the author of the thesis assigned this problem to students between ages 11 and 18 and evaluated their approaches. This problem is probably interesting for the people studying history of mathematics, too. However, the problem seems as a typical packing problem, so it is interesting also from the viewpoint of pure mathematics.

It can be shown simply by calculating the area of the triangle that the upper bound for the number of houses is 20. In the paper Nikolai Yu. Zolotykh: Alcuin's Propositiones de Civitatibus: the Earliest Packing Problems, arxiv:1308.0892, it is shown that it is possible to place 16 rectangles into the triangle.

I would like to know whether this problem was studied elsewhere. Are some better upper bounds on the number of rectangles known? (Or perhaps even an optimal solution?)


The best resource for "practical" packing problems I know of is Eric Friedmans' "Packing Palace." He has an answer to this problem for equilateral triangles and squares. Sadly, he does not address rectangles in triangles.


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