Magic Triangles

• Published in 2022
• Added on 2026-01-12

Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of integers with common sums in three directions, which we call magic triangles. For small sizes of these triangles, we count the number of unique magic triangles and examine distributions of integers at different positions within them. While we cannot enumerate the number of magic triangles at larger sizes, we offer a simulated annealing method for finding magic triangles.

Links

• https://arxiv.org/abs/2208.12577v1
• https://arxiv.org/pdf/2208.12577v1

Other information

BibTeX entry

@article{MagicTriangles,
	key = {MagicTriangles},
	type = {article},
	title = {Magic Triangles},
	author = {Gabriel Hale and Bjorn Vogen and Matthew Wright},
	abstract = {Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of integers with common sums in three directions, which we call magic triangles. For small sizes of these triangles, we count the number of unique magic triangles and examine distributions of integers at different positions within them. While we cannot enumerate the number of magic triangles at larger sizes, we offer a simulated annealing method for finding magic triangles.},
	comment = {},
	date_added = {2026-01-12},
	date_published = {2022-01-12},
	urls = {https://arxiv.org/abs/2208.12577v1,https://arxiv.org/pdf/2208.12577v1},
	collections = {easily-explained,fun-maths-facts,integerology},
	url = {https://arxiv.org/abs/2208.12577v1 https://arxiv.org/pdf/2208.12577v1},
	urldate = {2026-01-12},
	year = 2022,
	archivePrefix = {arXiv},
	eprint = {2208.12577},
	primaryClass = {math.GM}
}