Answer to August 30, 2004 Problem

The Kate and Jackson Problem

 
This week's Problem is in honor of our newest grandchildren who were born this summer -- Kate Cotter and Jackson Harding. Given the set of letters below, you are to spell the names KATE and JACKSON 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 names KATE and JACKSON by moving from the top letter and ending at the bottom letter?

          K          
        A   A        
      T   T   T      
    E   E   E   E    
  J   J   J   J   J  
A   A   A   A   A   A
  C   C   C   C   C  
    K   K   K   K    
      S   S   S      
        O   O        
          N          

 

Solution to the Problem:

There are 252 different ways to spell the names. You could use "Pascal's Triangle" to help solve it. I have replaced the letters of KATE and JACKSON by numbers representing how many different ways you could reach that particular letter.

I also accepted separate answers for each: there are 8 ways to spell Kate and 62 ways to spell Jackson (using Pascal's Triangle).

          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. Keith Mealy * Cincinnati, Ohio
2. Richard Johnson LaJolla, California
3. James Alarie University of Michigan -- Flint,
Flint, Michigan
4. Jeffrey Gaither Winchester, Virginia
5. David & Judy Dixon Bennettsville, South Carolina
6. David Amos Winchester, Virginia
7. Larry Schwartz Trumbull, Connecticut
* Keith became a grandfather for the first time this summer. His daughter Renata had twin boys!