**THE TORN NUMBER.— solution**

The other number that answers all the requirements of the puzzle is 9,801. If we divide this in the middle into two numbers and add them together we get 99, which, multiplied by itself, produces 9,801. It is true that 2,025 may be treated in the same way, only this number is excluded by the condition which requires that no two figures should be alike.

The general solution is curious. Call the number of figures in each half of
the torn label n. Then, if we add 1 to each of the exponents of the prime
factors (other than 3) of 10^{n} - 1 (1 being regarded as a
factor with the constant exponent, 1), their product will be the number of
solutions. Thus, for a label of six figures, n = 3. The factors of
10^{n} - 1 are 1^{1} × 37^{1} (not
considering the 3^{3}), and the product of
2 × 2 = 4, the number of solutions. This always includes the
special cases 98 - 01, 00 - 01, 998 - 01,
000 - 001, etc. The solutions are obtained as follows:—Factorize
10^{3} - 1 in all possible ways, always keeping the powers of
3 together, thus, 37 × 27, 999 × 1. Then solve the equation
37x = 27y + 1. Here x = 19 and y = 26.
Therefore, 19 × 37 = 703, the square of which gives one
label, 494,209. A complementary solution (through
27x = 37x + 1) can at once be found by
10^{n} - 703 = 297, the square of which gives
088,209 for second label. (These non-significant noughts to the left must be
included, though they lead to peculiar cases like 00238 - 04641 =
4879^{2}, where 0238 - 4641 would not work.) The special case
999 × 1 we can write at once 998,001, according to the law shown
above, by adding nines on one half and noughts on the other, and its
complementary will be 1 preceded by five noughts, or 000001. Thus we get the
squares of 999 and 1. These are the four solutions.

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