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## 12-digit Oddities

Math brain teasers require computations to solve.

 Puzzle ID: #43854 Fun: (2.47) Difficulty: (3.2) Category: Math Submitted By: javaguru

What characteristic do these three 12-digit numbers share with each other, but with no other 12-digit number?

100307124369
111824028801
433800063225

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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