# An Arithmetic loving Ant crawls to one Hundred

This ant can do arithmetic but can crawl only horizontally or vertically, never diagonally.

It starts from one of the cells shown in the picture below. It’s path covers thirteen different numbers that total exactly one hundred.

Can you trace the sequence of numbers it has visited in order?

The grid contains all integers from $$1$$ to $$15$$ except $$11$$. If we add up these $$14$$ unique integers we get $$109$$. Therefore, if we are to have $$13$$ distinct numbers add up to $$100$$ then the only number which will not appear in the path is $$9$$. So, we have to construct a path which avoids $$9$$ but gets every other unique number.
The numbers $$12$$, $$14$$, and $$8$$ each just appear once in the grid so they become like linchpins of the path. Notice that if the path were to proceed upward from $$12$$, the eliminated numbers force a path downward from $$14$$, inevitably to join with $$12$$. Then, the traversal of the $$1$$ and $$15$$ lower down make the capture of the $$8$$ and a $$10$$ in the upper part of the grid difficult (impossible, I think). This is why we begin going down from $$12$$. From there, it makes sense to avoid the lower $$13$$ so we can get through to the $$8$$ later on and the path is essentially forced after that.