\( 1 \leftarrow \quad \) How many of the numbers from 10 through 86 have the sum of their digits equal to a perfect square? There are \( \square \) numbers whose digits sum to a perfect square.

To find how many numbers from 10 through 86 have the sum of their digits equal to a perfect square, we first identify the potential perfect squares in this range. The perfect squares less than 18 (the maximum possible sum of the digits for the numbers in this range) are 1, 4, 9, and 16. Now, we analyze the numbers from 10 to 86: - The digits for tens place range from 1 to 8 and for the ones place from 0 to 9. - The sum of digits for a two-digit number \( ab \) (where \( a \) is the tens digit and \( b \) is the ones digit) is given by \( a + b \). Here's the breakdown: 1. **For each tens digit (1 to 8)**: - **Tens digit 1**: Possible ones digits are 0 through 9, sums range from 1 to 10. - **Tens digit 2**: Possible ones digits are 0 through 9, sums range from 2 to 11. - **Tens digit 3**: Possible ones digits are 0 through 9, sums range from 3 to 12. - **Tens digit 4**: Possible ones digits are 0 through 9, sums range from 4 to 13. - **Tens digit 5**: Possible ones digits are 0 through 9, sums range from 5 to 14. - **Tens digit 6**: Possible ones digits are 0 through 9, sums range from 6 to 15. - **Tens digit 7**: Possible ones digits are 0 through 9, sums range from 7 to 16. - **Tens digit 8**: Possible ones digits are 0 through 6, sums range from 8 to 14. 2. **Identify combinations that sum to perfect squares (1, 4, 9, 16)**: - For 1: Only the number 10. - For 4: Numbers 13, 22, 31, 40. - For 9: Numbers 18, 27, 36, 45, 54, 63, 72, 81. - For 16: Numbers 79, 88 (but only 79 is valid since we stop at 86). 3. **Counting**: - **Sum = 1**: 1 number (10) - **Sum = 4**: 4 numbers (13, 22, 31, 40) - **Sum = 9**: 8 numbers (18, 27, 36, 45, 54, 63, 72, 81) - **Sum = 16**: 1 number (79) Adding them gives us: \( 1 + 4 + 8 + 1 = 14 \). Thus, there are \( \boxed{14} \) numbers whose digits sum to a perfect square from 10 to 86.