12digit Oddities
Math brain teasers require computations to solve.
What characteristic do these three 12digit numbers share with each other, but with no other 12digit number?
100307124369
111824028801
433800063225
Hint316713
334401
656635
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Answer
They are all square numbers:
100307124369 = 316713^2
111824028801 = 334401^2
433800063225 = 656635^2
The sum of their digits are square numbers:
1+0+0+3+0+7+1+2+4+3+6+9 = 36 = 6^2
1+1+1+8+2+4+0+2+8+8+0+1 = 36 = 6^2
4+3+3+8+0+0+0+6+3+2+2+5 = 36 = 6^2
The sum of their digit pairs are square numbers:
10+03+07+12+43+69 = 144 = 12^2
11+18+24+02+88+01 = 144 = 12^2
43+38+00+06+32+25 = 144 = 12^2
The sum of their digit triplets are square numbers:
100+307+124+369 = 900 = 30^2
111+824+028+801 = 1764 = 42^2
433+800+063+225 = 1521 = 39^2
The sum of their digit quadruplets are square numbers:
1003+0712+4369 = 6084 = 78^2
1118+2402+8801 = 12321 = 111^2
4338+0006+3225 = 7569 = 87^2
The sum of their digit sextuplets are square numbers:
100307+124369 = 224676 = 474^2
111824+028801 = 140625 = 375^2
433800+063225 = 497025 = 705^2
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Comments
markmonnin
Jan 07, 2009
 That's amazing trivia! 
javaguru
Jan 07, 2009
 Thanks! I've never seen these before, but I was wondering if there were any that fit this pattern, so I wrote a program to find them.
Since then I submitted several series to the Online Encyclopedia of Integer Sequences (OEIS) about these numbers. (See http:// www.research.att.com/ ~njas/sequences/A153745.) 
SweetiePi
Feb 15, 2009
 Wow. Cool! 
Jimbo
Mar 29, 2009
 Very interesting but you surely aren't expecting us to get all of those properties without looking at the answer , are you? 
javaguru
Mar 29, 2009
 I agree this is more of a puzzle than a teaser, but I think it's possible to get the answer. If you observe that the numbers are all squares and that their digital sums are all squares, I think it's possible to get the next step with the sum of the digit pairs. 
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