I'm not sure what Davis is `seeing' but perhaps not all of: Andrew Bremner \& Richard K.~Guy, The delta-lambda configurations in tiling the square, {\it J.~Number Theory}, {\bf 32}(1989) 263--280; {\it MR} {\bf 90g}:11031. Andrew Bremner \& Richard K.~Guy, Nu-configurations in tiling the square, {\it Math.\ Comput.}, {\bf59}(1992) 195--202, S1--S20; {\it MR} {\bf93a}:11019. Richard K.~Guy, Tiling the square with rational triangles, in R.~A.~Mollin (editor) Number Theory \& Applications, ({\it Proc.\ N.A.T.O.\ Adv.\ Study Inst.}, Banff 1988), Kluwer, Dordrecht, 45--101; {\it MR} {\bf92f}:11044. What we'd REALLY like to know is: Is there a point at integer distances from each corner of a square with integer edge? Best to all, R. On Fri, 29 Apr 2005, David Wilson wrote:
In the Mathworld article "Square Dissections" we find
Guy (1989) asks if it is possible to triangulate a square with integer side lengths such that the resulting triangles have integer side lengths (Trott 2004, p. 104).
There's more to this than I'm seeing right?
If I interpret this to mean dissecting an square into integer-sided triangles, it doesn't seem very difficult. I can easily dissect a 12x12 square by tiling it with 3x4 rectangles and cutting the rectangles along random diagonals. I played with removing edges from those tilings and got a tiling of the 12x12 in 5 integer-sided triangles: 3-4-5, 5-5-6, 6-8-10, 10-10-12 and 9-12-15. But that solution dissection seems too simple that Guy would have overlooked it, so I'm thinking the original problem was more involved.
So now the question is, can a square be dissected into fewer than 5 integer-sided triangles?
1 and 2 are clearly out. For 3, we cant use the square diagonal, so one of the triangles must have a square edge as a base and apex interior to the opposite side. I couldn't find integer solutions to this.
For 4 triangles, I could not be sure I got all the possible configurations, but it might be an interesting investigation.
- David W. Wilson
"Truth is just truth -- You can't have opinions about the truth." - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"
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