The numbers 1 and 44 form squares when they are concatenated in either order, namely 144=12^2 and 441=21^2.
Are there 3 integers A, B and C such that all 6 of their possible concatenations form squares?
If not, what is the largest number of squares that can be formed by concatenating 3 integers?
I tested integers up to 1000, but I found no solutions if A, B, and C must be distinct.
I did find 9 solutions in which one of the numbers appears twice. In each of these instances there were 4 of the 6 permutations resulting in a square (but only 2 unique squares).
In other words, for the pattern A, B, B (for which there are 3 possible permutations rather than 6), in these solutions, 2 of the 3 permutations are squares.
(A, B, B)
[squares]
(1, 1, 664)
[11664, 16641]
(1, 4, 4)
[144, 441]
(16, 64, 64)
[166464, 646416]
(100, 400, 400)
[100400400, 400400100]
(121, 484, 484)
[484484121, 121484484]
(144, 576, 576)
[144576576, 576576144]
(169, 676, 676)
[676676169, 169676676]
(196, 784, 784)
[196784784, 784784196]
(225, 900, 900)
[900900225, 225900900]
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Posted by Larry
on 2024-05-26 09:14:39 |
Some findings
