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12digit Oddities
Category:  Math 
Submitted By:  javaguru 
Fun:  (2.44) 
Difficulty:  (3.22) 
What characteristic do these three 12digit numbers share with each other, but with no other 12digit number?
100307124369
111824028801
433800063225
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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