I recently wrote my friend a birthday card and thought it would be fun to write her age using mathematical operations on the digits of her birth-year in order. For example she turned 36 and was born in 1976 so I wrote:
Happy $ (-(1) ^9 + 7)\times 6 $ th birthday.
The question is for which birth-year and age combinations is this not possible?
I will allow the following operations: addition/subtraction, multiplication/division, exponentiation, factorials, square root (without requiring a 2, higher roots will require digits) and $log_{10}(x)$. In order to limit the size of the problem, consider birth-years from 1950 and ages up to 62. Aim is to obtain lists as in the following example:
Born in 1950 age of 1: $((1+9)/5) - 0!$
Born in 1950 age of 2: $((1+9)/5) \times 0!$
...
Born in 1950 age of 60: $((1+9) \times (5+ 0!)$
Born in 1950 age of 61: ?
Born in 1950 age of 62: ?
Remember - solutions must include all the digits of the year in order. Have fun!