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How far can you go? (Posted on 2015-05-12) Difficulty: 2 of 5
16 is the smallest 2-digit square whose digits are non-decreasing.
144 is the smallest 3-digit square whose digits are non-decreasing.
1156 is the smallest 4-digit square whose digits are non-decreasing.

GO ON!

No Solution Yet Submitted by Ady TZIDON    
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Solution Go on, indeed. | Comment 1 of 6
You might think that beyond a certain length, the sequence must end but it does not.

Although not always the smallest, note the sequences:
34^2=1156 
334^2=111556
3334^2=11115556
which covers all the even lengths except 2
17^2=289
167^2=27889
1667^2=2778889
which covers all the odd lengths

The above are sometimes, but not always, the smallest.
The actual sequence begins
16 = 4^2
144 = 12^2
1156 = 34^2
11236 = 106^2
111556 = 334^2
2666689 = 1633^2
11115556 = 3334^2
277788889 = 16667^2
1111155556 = 33334^2
11122233444 = 105462^2
the next few terms use the sequences I gave at the top.

see https://oeis.org/A028820/b028820.txt
which lists all squares with increasing terms but is easy to scan for the smallest of each length.

  Posted by Jer on 2015-05-12 09:36:57
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