Concatenated squares are squares that can be formed by stringing other squares together. So instead of adding, the squares are treated as strings and concatenated. For example, 49 \((7^2)\) is the smallest concatenated square, consisting of 4 \((2^2)\) and 9 \((3^2)\) strung together:

\(49 = 7^2 = 2^2~\&~3^2\)

I use the ampersand (&) to indicate concatenation.

Once you start looking for them, you can find lots of concatenated squares:

\( 100 = 1 ~\&~ 0 ~\&~ 0 \)

\( 144 = 1 ~\&~ 4 ~\&~ 4 \)

\( 169 = 16 ~\&~ 9 \)

\( 361 = 36 ~\&~ 1 \)

\( 400 = 4 ~\&~ 0 ~\&~ 0 \)

Of course, the ones ending in “00” are not very interesting. They are trivial.

Squareness is an immutable characteristic of an integer; it does not depend on number base. Thus \(9\) is a square whether it is written as \(1001_{(2)}\), \(21_{(4)}\), \(11_{(8)}\), or \(9_{(10)}\). Concatenation, however, is number base dependent. So, for example, all squares can be expressed as concatenated squares in binary. We are only concerned here with decimal squares.

Some special concatenated squares can also be expressed as the sum of squares (i. e., as Pythagorean triples). For example, \(1681 ~(41^2) = 16 ~\&~ 81\) and \(1681 ~(41^2) = 1600 ~(40^2) + 81 ~(9^2)\). \(1681\) is the only four-digit example of what we may call a Pythagorean concatenated square. (Of course, other concatenated squares may be expressed as the sum of squares without the elegance and symmetry of \(1681\). For example, \(169 = 16 ~\&~ 9\) and \(169 = 144 + 25\).)