Try to consider the problem of the youngest Alice in such way: make each grandchild has a similar age as possible, and the age of each grandchild should be different, that is, $$1 + 2 + 3 +\cdots 9 + 10 = 55$$; then $$180 - 55 = 125$$, $$125 \div 10 = 12R5$$, and $$5$$ is left. If every child\textquotesingle s age is added by $$12$$, then $$5$$ is left. Give the extra year to each of the $$5$$ oldest children. In doing so, the minimum age of Alice is $$10+12+1=23$$. 
Granny has $$10$$ grandchildren. Alice is the oldest. One day, Granny notices that her grandchildren all have different ages. If the sum of her grandchildren\textquotesingle s ages is $$180$$, what is the youngest Alice can be? (2014 Math Kangaroo Problem, Level 5-6, Question \#30) 

[a] $$19$$ 
[b] $$20$$ 
[c] $$21$$ 
[d] $$22$$ 
[e] $$23$$ 