Answer to the Problem of the Week

for the week of March 6, 2000

MATHEMATICS Problem
Given the set of letters below, you are to spell the word MATHEMATICS by starting at the top of the rhombus and moving down through the letters. At each level, you must move to a letter which is below and to the immediate left or below and to the immediate right of the letter you are at. In how many ways can you spell out the word mathematics by moving from the top letter and ending at the bottom letter? 
                   M
                 A   A 
               T   T   T
             H   H   H   H
           E   E   E   E   E
         M   M   M   M   M   M
           A   A   A   A   A
             T   T   T   T
               I   I   I
                 C   C
                   S


  Here is one possible way to spell out
  mathematics:

                   M
                    \
                 A   A
                    /
               T   T   T
                    \
             H   H   H   H
                      \ 
           E   E   E   E   E
                        \                         
         M   M   M   M   M   M
                        / 
           A   A   A   A   A
                      /
             T   T   T   T
                    /
               I   I   I
                  /
                 C   C
                  \
                   S


Solution:

The answer is 252 ways!

You could use "Pascal's Triangle" to help solve it. I have replaced the letters of mathematics by numbers representing how many different ways you could reach that particular letter. 

                     1
                   1   1 
                 1   2   1
               1   3   3   1
             1   4   6   4   1
           1   5  10  10   5   1
             6   15  20  15  6
               21  35  35  21
                 56  70  56
                  126  126
                    252

Correctly solved by:

1. Jon Pence Winchester, VA