I went to a birthday party yesterday for my mom and my two nephews. They don’t share the same birthday, but it’s pretty close. Each of the boys brought a candle in the shape of their age (one looked like a five, the other a seven). They put their candles on the cake and the resulting number was my mom’s age (57). We all stared in wonderment at the mathematical anomaly and posited that it would happen again in some number of years.
Of course it wouldn’t since the combination of the boys’ ages would increase 10 years for every one year my mom ages. The current situation can be described as
x + 52 = 10x + x + 2 where x is the junior boy’s age. I wanted to determine at what point this situation would hold if the boys’ ages were switched, that is the candles were arranged to show 75 instead of 57. With the senior boy’s age as x,
x + 50 = 10x + x - 2, or
x=5.2. When the senior boy was 5.2 years old, the junior boy was 3.2 years old and my mom was 55.2 years old. Not the awe inspiring numbers as in the first example.
Next, I tried it with another adult who turned 38 on about the same day:
x + 33 = 10x + x + 2, which results in the junior, senior, and adult being 3.1, 5.1, and 36.1, respectively. Setting the senior boy’s age as x resulted in another non-integer solution.
I concluded that difference between the adult and junior boy had to end in the same number as the difference between the senior and junior. That is, if the boys are three years apart, the difference between the adult and the junior would have to end in a 3 (13, 23, 33, 43, etc…). Similarly, if the senior boy’s age was put first, the difference between the adult and the senior boy had to end in a number that was ten minus the difference between the senior and the junior. That is, if the boys are four years apart, the difference between the adult and the senior would have to end in 6.
It then dawned on me that the only way either of these situations could happen is if the adult’s age was evenly divisible by ten when one of the boys was born. In the first example, my mom was 50 when senior was born. No matter when junior was born, the difference between my mom’s age and junior’s age would always end in the same number as the difference between the boys’ ages. Had junior been born 1 year later, the difference between my mom and junior would have been 51 (51-0) which ends in 1 – the difference between the boys’ ages. In that case, when junior was five, senior would be six, and adult would be 56.
Conversely, for the situation to work using the senior boy’s age first, the adult’s age would have to be evenly divisible by 10 when junior was born. Had my mom been 50 when junior was born, then senior’s fifth birthday would have resulted in senior, junior, adult being five, three, and 53, respectively.
Conclusion: If you want to save money on candles in one year when three people in your family have close birthdays, have one of the kids in a year when the adult’s age is divisible evenly by 10. And please don’t spend as much time thinking about this as I have.