A father's age today is a two-digit number, and his son's age today is that same number with its two digits interchanged. The father is 36 years older than the son, and the two digits add up to 10. How old is the father?

Write the father as $10t+u$ and the son as $10u+t$. Their difference is $(10t+u)-(10u+t)=9(t-u)$, so $9(t-u)=36$ and $t-u=4$. With $t+u=10$ as well, $t=7$ and $u=3$, making the father 73 and the son 37. 37 is the son's age - the right pair of digits, the wrong person. 82 reads the difference straight off as $t-u=6$ and misses the factor of 9; 82 and 28 differ by 54, not 36. 91 also has digits adding to 10, but 91 and 19 differ by 72.