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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!

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2  
Do you think that $\log_{10}(0)=1$?? It is $-\infty$ actually. –  Aang Sep 27 '12 at 11:55
2  
you can use $0!$ instead. –  Aang Sep 27 '12 at 12:26
    
Whoops, my mistake! Now edited to be correct. –  Epictetus Sep 27 '12 at 14:09
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