Monday, May 19, 2008

Google Treasure Hunt 2008

So Google is running a treasure hunt. 1 puzzle a week for 4 weeks. I love puzzles.

The first puzzle they have up goes something like this:
A robot is located at the top-left corner of a 52 x 52 grid. The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid. How many possible unique paths are there?

For a while I tried to picture the mechanics of this problem in my head. It was too much,.. I would soon get lost trying to keep track of a dozen different paths and shapes all at once. So i resorted to paper. Not too long and I was having good progress. I would draw grids with numbers in each cell representing how many paths could be made from that cell to the bottom right. The trick was to start small.

I drew grids for 2x2 and 3x3. I still couldnt work out the pattern,.. Then I drew 2x3 and 3x2 grids; Bingo.

Overlay/Imageine a 3x2 and 2x3 grid at the bottom-right of the grid above; To work out the total number of paths from the last missing cell (of 3x3) is simply all paths available in both the 2x3 and 3x2 grids. Still dont understand,..
Simply put: The number of unique paths from any cell to the finish is equal to the sum of, the number of paths from the cell on the right, and the cell below it.

Here is the Java source I used to calculate my grid (55 x 60). My answer was 696940125414123253093858308567840.

I would be interested in looking over the mathematical formula that calculates this directly incase anyone happens to know it.

Happy treasure hunting!

PS: My code calculates the map upside down and other way around... In other words, the robot is going from bottom right to top left. Not the same as the picture above.
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