Answer to Problem of the Week for 04/02/2001

Answer to April 2, 2001 Problem

from Student Math Notes March 1985

The Pascal Problem

Given the set of letters below, you are to spell the word PASCAL by starting at the center of the rhombus and moving through the letters. Each move from center must be to the right, left, up or down (in other words, you may not move diagonally). How many ways can you find to spell the word PASCAL? L L A L L A C A L L A C S C A L L A C S A S C A L L A C S A P A S C A L L A C S A S C A L L A C S C A L L A C A L L A L L Here is one possible way to spell out PASCAL: L L A L L A C A L L A C S C A L \| L A C S A S--C--A L \| L A C S A P--A S C A L L A C S A S C A L L A C S C A L L A C A L L A L L
Solution to Problem: The answer is 124 ways! You could use "Pascal's Triangle" to help solve it. I have replaced the letters of PASCAL by numbers representing how many different ways you could reach that particular letter. Then add up all the numbers where the letter L is (all the numbers around the outside of the figure) to get your final answer. 1 5 1 5 10 4 1 4 10 10 6 3 1 3 6 10 5 4 3 2 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 5 4 3 2 1 2 3 4 5 10 6 3 1 3 6 10 10 4 1 4 10 5 1 5 1

Given the set of letters below, you are to spell the word PASCAL by starting at the center of the rhombus and moving through the letters. Each move from center must be to the right, left, up or down (in other words, you may not move diagonally). How many ways can you find to spell the word PASCAL?

                   L
                 L A L 
               L A C A L
             L A C S C A L
           L A C S A S C A L
         L A C S A P A S C A L
           L A C S A S C A L
             L A C S C A L
               L A C A L
                 L A L 
                   L


  Here is one possible way to spell out
  PASCAL:

                    L

                 L  A  L 
 
              L  A  C  A  L

           L  A  C  S  C  A  L
                             |
        L  A  C  S  A  S--C--A  L
                       |
     L  A  C  S  A  P--A  S  C  A  L

        L  A  C  S  A  S  C  A  L

           L  A  C  S  C  A  L

              L  A  C  A  L

                 L  A  L 

                    L

Solution to Problem:

The answer is 124 ways!

You could use "Pascal's Triangle" to help solve it. I have replaced the letters of PASCAL by numbers representing how many different ways you could reach that particular letter. Then add up all the numbers where the letter L is (all the numbers around the outside of the figure) to get your final answer.

                    1

                 5  1  5 
 
             10  4  1  4  10

          10  6  3  1  3  6  10
                              
        5  4  3  2  1  2  3  4  5
                         
     1  1  1  1  1  1  1  1  1  1  1

        5  4  3  2  1  2  3  4  5

          10  6  3  1  3  6  10

             10  4  1  4  10

                 5  1  5 

                    1

Correctly solved by:

1. Walt Arrison	Philadelphia, Pennsylvania
2. Rick Jones	Kennett Square, Pennsylvania
3. George Gaither	Winchester, Virginia
4. Chip Schweikarth	Winchester, Virginia
5. James Alarie	University of Michigan -- Flint
6. Keith Mealy	Cincinnati, Ohio
7. Erin McGinnis	Winchester, Virginia
8. Gusti Oggenfuss	Montet, Switzerland