How many two-digit numbers have digits whose sum is a perfect square?
Well, first what is the greatest sum you can have? This would occur if the number was 99 and the answer would be 18. This means we want perfect squares less then 18. The only perfect squares that we could have is 1, 4, 9, and 16. The only number that could sum to equal 1: 10 The numbers to equal 4: 13, 22, 31, 40 Numbers to equal 9: 18, 27, 36, 45, 54, 63, 72, 81, 90 Numbers to equal 16: 79, 88, 97 This makes our list: 10, 13, 18, 22, 27, 31, 36, 40, 45, 54, 63, 72, 79, 81, 88, 90, 97. This brings our total to: 17 total. If you break problems like this up systematically, it becomes easier. For instance, pick each number as shown above. Now, pick the smallest number that you could use for the first digit. This will be 1 for all numbers less then 10. For numbers greater then 10, set the one’s digit to 9 and find the ten’s digit to make it work. Next, increase the ten’s digit by one increment and decrease the one’s digit by one. This ensures that the sum doesn’t change. After one list is
