You are allowed to use +, -, / and * (plus, minus, division and multiplication) signs and bracketing. These signs you can put between the numbers
1963
to form mathematical expressions. You must put at least one sign between two numbers and – cannot be used as “negative”, thus -1+9+6+3 is not allowed, but 1-9+6+3 is allowed.
The question is: what is the smallest natural number that cannot be expressed in this way? How to show (elegantly, without computer) that it is impossible to get 14?
I found myself the following representations:
$0=1*9-6-3$
$1=(1/9)*(6+3)$
$2=1+9/(6+3)$
$3=1*9/(6-3)$
$4=1+9/(6-3)$
$5=(1+9)/(6/3)$
$6=1*9-6+3$
$7=1+9-6+3$
$8=1+9-6/3$
$9=1*(9-6)*3$
$10=1+(9-6)*3$
$11=1*9+6/3$
$12=1+9+6/3$
$13=1+9+6-3$