IBM Research | Ponder This | July 2011 solutions

# Ponder This

## July 2011

<<June July August>>

We can solve the problem for 81 = 3^4 students using linear codes. Write each student as a 4-digit number in base 3, and let him eat the scalar multiplication of his number with the following constants:

(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (0,0,1,1), (0,0,1,2), (1,1,0,0), (1,2,0,0), (0,1,0,1), (0,1,0,2), (1,0,1,0), (1,0,2,0), (1,1,1,1), (1,1,2,2)

And click here to see another representation sent to us by Gale Greenlee.

14 is the smallest number of days we found with a solution, yet we can prove a lower bound of 10. We leave this gap as an open question for our readers.

This month James Dow Allen solved the unasked bonus question: Why did we ask for lines 80 characters in length when the solution lends itself to 81? Of course, we did it to make the solution a little less obvious, but also to honor the 80-character-wide IBM punched card.

# Here it is in the format we requested:

ABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCAB

AAABBBCCCAAABBBCCCAAABBBCCCAAABBBCCCAAABBBCCCAAABBBCCCAAABBBCCCAAABBBCCCAAABBBCC

AAAAAAAAABBBBBBBBBCCCCCCCCCAAAAAAAAABBBBBBBBBCCCCCCCCCAAAAAAAAABBBBBBBBBCCCCCCCC

AAAAAAAAAAAAAAAAAAAAAAAAAAABBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCC

AAAAAAAAABBBBBBBBBCCCCCCCCCBBBBBBBBBCCCCCCCCCAAAAAAAAACCCCCCCCCAAAAAAAAABBBBBBBB

AAAAAAAAABBBBBBBBBCCCCCCCCCCCCCCCCCCAAAAAAAAABBBBBBBBBBBBBBBBBBCCCCCCCCCAAAAAAAA

ABCBCACABABCBCACABABCBCACABABCBCACABABCBCACABABCBCACABABCBCACABABCBCACABABCBCACA

ABCCABBCAABCCABBCAABCCABBCAABCCABBCAABCCABBCAABCCABBCAABCCABBCAABCCABBCAABCCABBC

AAABBBCCCAAABBBCCCAAABBBCCCBBBCCCAAABBBCCCAAABBBCCCAAACCCAAABBBCCCAAABBBCCCAAABB

AAABBBCCCAAABBBCCCAAABBBCCCCCCAAABBBCCCAAABBBCCCAAABBBBBBCCCAAABBBCCCAAABBBCCCAA

ABCABCABCBCABCABCACABCABCABABCABCABCBCABCABCACABCABCABABCABCABCBCABCABCACABCABCA

ABCABCABCCABCABCABBCABCABCAABCABCABCCABCABCABBCABCABCAABCABCABCCABCABCABBCABCABC

ABCBCACABBCACABABCCABABCBCABCACABABCCABABCBCAABCBCACABCABABCBCAABCBCACABBCACABAB

ABCBCACABCABABCBCABCACABABCCABABCBCABCACABABCABCBCACABBCACABABCABCBCACABCABABCBC

# Representation sent to us by Gale Greenlee

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com