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 Neither 3 nor 6 (Posted on 2011-03-11)
0, 25, 2025, 13225…are squares that remain squares if every digit in the number defining them is augmented by 1.
Let's call them squarish numbers.

a. List two more samples of squarish numbers.
b. Prove that all such numbers are evenly divisible by 25.
c. Why are there neither 3-digit nor 6-digit squarish numbers?
d. Prove that between 10^k and 10^(k+1) there is at most one squarish number.

 See The Solution Submitted by Ady TZIDON No Rating

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 Part a only (spoiler) | Comment 1 of 6

Here are three more added to the list:

`squarish     sqrt          augmented      sqrt                             square25           5              36            62025         45             3136          5613225        115            24336         1564862025      2205           5973136       244460415182025  245795         71526293136   267444207612366025 455645         318723477136  564556`

It seems lengths 8, 9 and 10 are missing also.

DEFDBL A-Z

repu = 1: rep9 = 9 * repu
FOR n = 1 TO 9999999
sq = n * n
WHILE sq > rep9
repu = repu * 10 + 1
rep9 = 9 * repu
WEND
s\$ = LTRIM\$(STR\$(sq))
IF INSTR(s\$, "9") = 0 THEN
tst = sq + repu
sr = INT(SQR(tst) + .5)
IF sr * sr = tst THEN
PRINT sq; n, tst; sr
END IF
END IF
NEXT n

 Posted by Charlie on 2011-03-11 13:42:19

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