### Brain Teasers

# 12-digit Oddities

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

100307124369

111824028801

433800063225

100307124369

111824028801

433800063225

### Hint

316713334401

656635

### 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

That's amazing trivia!

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.)

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.)

Wow. Cool!

Very interesting but you surely aren't expecting us to get all of those properties without looking at the answer , are you?

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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