Peter Nugent Project Euler
Problem 15- Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner. How many such routes are there through a 20×20 grid? (96463 solved)
To solve this problem, I represented the paths as a series of R’s and D’s, representing right and down steps. For a K×K grid, the path must be 2K long, containing K R’s and K D’s. You can solve this using a choose function to find all the valid combinations of R’s and D’s. For a 20×20 grid, it will be a 40 step path with 20 R’s and 20 D’s, so we can find them all by performing the function 40 choose 20. This gives a number that would be impractical to calculate by hand, so I used Wolfram Alpha to find the solution.
Solution- 137846528820

Problem 16- 215 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26. What is the sum of the digits of the number 21000? (120664 solved)
This problem is fairly straightforward, but the value 21000 is far too large to do by hand or even a regular calculator. To solve this, I used Wolfram Alpha to find 21000. Afterwards, I also used Wolfram Alpha to find the sum of its digits, even though it would have been possible to do by hand or with a less powerful calculator.
Solution- 1366

Problem 18- If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total. If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would be used? (79465 solved)
To solve this problem, I had to use some code someone else had already written. I found it on a Microsoft support site discussing changing numbers from numerals to words on Excel. I found the code I needed here - http://support.microsoft.com/kb/213360. I used this for the numbers 1-1000. After that, I used the character count function in Excel, and subtracted the amount of spaces in each number, as they are not counted in our solution, but are counted by the character counter. Finally, I used a sum on the cells with the character counters to arrive at our final solution. Without excel, this would have been doable, but much more time consuming and with much more room for error.
Solution- 21124