Palindromic Ages

Tom Smith is 86 years old and his daughter, Laura, is 59 years old Tom Smith is 86 years old and his daughter, Laura, is 59 years old. One day he was looking through his family photo albums and came upon an old photo of himself with his daughter, Laura. He looked at the date which had been scribbled on the back of the photo and realised that the photo was taken when he was 41 years old and Laura was 14 years old. ‘Just a minute!’ he thought, ‘ Laura’s age was the reverse of my age. Our ages were palindromic!!!’ This made Tom think a little more and he looked through more photo albums. He realised that this had happened a few times as they both grew older.

Tom also found a photo of himself with his son, Steven Tom also found a photo of himself with his son, Steven. He looked at the date of this photo and realised that this photo was taken when he was 45 years old and his son was 13 years old. He then wondered if their two ages had ever been palindromic? This made Tom think a little more and he continued looking through his photos. He realised that, over the years, his age and Steven’s age had never been palindromic.

Tom also found a photo of himself with his brother, John, when they were 40 and 31 years old respectively. Were there ages ever palindromic? Another photo showed him with his sister, June, when they were 38 and 36 years old respectively. Were there ages ever palindromic?

Can you help him with the mathematics? Tom tried to work out why it worked with some members of his family and not with others. Can you help him with the mathematics?