Consider the number 4808
Notice: 4^2+8^2=80
i.e. the sum of the squares of this number's first and last digits equals the number obtained when the first and last digits are erased.

How many numbers with such feature exist below 10000?

Clearly, no leading zeroes.

(In reply to Just counting (spoiler) by Steve Herman)

If the prohibition on leading zeros applies only to the 3- or 4-digit number itself, and not the internal 2-digit number, then there would be seven more, that is, 83, as the 1-digit  sums of squares would each account for both a 3-digit and a 4-digit number:

Edited on July 17, 2017, 9:57 pm

Posted by Charlie on 2017-07-17 13:53:40