If 11 x 11 =, 121 and 111 x 111 = 12,321, and 1,111 x 1,111 = 1,234,321 then, without using a calculator, what does 111,111,111 x 111,111,111 equal?

The product of the two numbers is a numeric palindrome; the correct answer is the one that reads the same backwards 12,345,678,987,654,321

In mathematical units the number can be expressed as:-

12 quadrillion,345 trillion,678 billion,987 million,654 thousand 321.

In multiplication table terms the proof can be demonstrated as:-

111111111 x 111111111

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000,000,000,111,111,111

000,000,001,111,111,110

000,000,011,111,111,100

000,000,111,111,111,000

000,001,111,111,110,000

000,011,111,111,100,000

000,111,111,111,000,000

001,111,111,110,000,000

011,111,111,100,000,000

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12345678987654321

Each digit of the result comes from summing the digits in one column -- that is, counting how many ones there are. Since this is at most 9, there are no carries between columns.