In the following square root extraction, write the appropriate digits in place of the dots

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The square of the first digit of the root is two-digit, its double is one-digit, so it can only be 4. The first subtrahend is 16. The second digit of the root is 1, because 1 is the only number that, when written after 8 and multiplied by the second digit of the root, gives a two-digit number. This implies that the first digit of the third row is 1, and the number under the root sign starts with 17. The fourth row is 81. The third digit of the root is a number that, when written after 82 and multiplied by the third digit of the root, gives a four-digit number whose second digit is $5-1=4$. The requirements are met by 3 and 9, and with this knowledge, the values of the still unknown digits are self-evident.

Thus, there are two solutions:

$$
\sqrt{170569}=413
$$

16

105: 81 × 1

\underline{81}

2469: 823 × 3

\underline{2469}

0

$$
\sqrt{175561}=419
$$

\underline{16}

155: 81 × 1

\underline{81}

7461: 829 × 9

\underline{7461}

0

Julius Kalmár (Hatvan, József Bajza Secondary School II. grade)