Student asked her maths teacher if he would reveal his age. Seeing an opportunity to test his student’s mental skills, the teach
er replied with a riddle: “My age in years is not a prime but is odd. Also, when the digits in my age are reversed and the new number added to my age, the result is a perfect square. Of course, if you prefer, you can take my age and reverse the digits and subtract that from my age and still get a perfect square.” Although 51 is odd and not a prime, the teacher cannot be 51 years old, because neither (51 + 15) nor (51 -15) are perfect squares, and they both need to be perfect squares.
If her age is 10 t + u (where t is the tens digit and u is the units digit) then reversing the digits gives 10 u + t. and the sum is 11 t + 11 u, which is a multiple of 11. We know this has to be a square number. Ong Xing Cong from Singapore sent in the following solution.
11 t + 11 u = 11 x 11 = 121
t + u = 11
65 - 56 = 3 x 3
She is 65 years old.
The best solutions do not need trial and improvement methods and they show that the answer or answers found are the only possible answers. Knowing the digits add up to 11 ( t + u = 11), you can also use (10 t + u ) - (10 u + t ) = 9 t - 9 u = 9( t - u ) As this is also a square number you know ( t - u ) is either 1, 4, or 9. The solutions for 4 and 9 don't give whole number values for t in 0  t  9.
