All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers > Sequences
Colored triangle (Posted on 2018-12-14) Difficulty: 4 of 5
Begin with a finite sequence of blocks in a row, each in one of 3 colors: red, blue, yellow.

Below each pair of neighboring blocks place a new block with the color rule: If the blocks are the same color use that color but if they are different use the third color.

Example:

r b y y b
 y r y r
  b b b
   b b
    b
How can the color of the last block be easily predicted from the top row?

Note: I don't know the full answer but can solve special cases.

No Solution Yet Submitted by Jer    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
proposed solution | Comment 5 of 10 |
Submitted without proof, as what I've come up with is really more observation than demonstration.

I modeled the three colors with 0,1,2 and noticed that for every pair (a,b) with resultant entry c, 3 divides (a+b+c).  So I tried a few calculations with the string abcdef... Looking at the first entry only gave:

Step 1   (a+b)
Step 2   (a+2b+c)
Step 3   (a+3b+3c+d)
Step 4   (a+4b+6c+4d+e)
Step 5   (a+5b+10c+10d+5e+f)

I know binomial coefficients when I see them, and since the first entry eventually becomes the final entry, the plan is clear how to evaluate a string with N entries.

1) find the coefficients of (x+y)^(N-1)
2) sum the products of the kth string value and the kth coefficient
3) subtract the result from the smallest multiple of 3 that is greater








 

 

  Posted by xdog on 2018-12-17 20:03:48
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (0)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information