**Source:**IBM Ponder This Dec 12 ( http://domino.research.ibm.com/comm/wwwr_ponder.nsf/challenges/December2012.html ) - Mailed to me by Aashay Harlalka (Final Year Student, CSE, IITB)

**Problem:**

36 people live in a 6x6 grid, and each one of them lives in a separate square of the grid. Each resident's neighbors are those who live in the squares that have a common edge with that resident's square.

Each resident of the grid is assigned a natural number N, such that if a person receives some N>1, then he or she must also have neighbors that have been assigned all of the numbers 1,2,...,N-1.

Find a configuration of the 36 neighbors where the sum of their numbers is at least 90.

As an example, the highest sum we can get in a 3x3 grid is 20:

1 2 1

4 3 4

2 1 2

Update (24 June 2014):

**Solution:**Available on IBM Research Ponder This - December 2012 Solution

3,2,1,2,3,1

ReplyDelete1,5,3,4,5,2

3,4,2,1,1,3

2,1,1,3,4,2

I don't think one can go beyond 86.

ReplyDelete3,1,1,2,2,1

2,5,4,3,5,1

1,3,2,1,4,2

2,4,1,2,3,1

2,5,3,4,5,1

1,1,2,1,2,3

well u ar ealmost right but just in the last row first coloumn and first row last coloumn u just need to change the 1's to 3's and u will get the total to be 90

DeleteNo, I can't do that because we need to have 1 in the neighbour of 2 (Second Last row, first column) and similarly at the other corner as well :-)

DeleteBut, the solution posted below is I think correct.

Well, I thing it could be this:

ReplyDelete1, 2, 2, 1, 3, 2

3, 5, 4, 3, 2, 1

2, 1, 1, 4, 5, 3

3, 5, 4, 2, 1, 2

1, 2, 3, 4, 5, 3

2, 3, 1, 1, 2, 1

1 2 3 2 3 1

ReplyDelete3 5 4 1 5 2

2 1 2 2 4 3

3 4 2 2 1 2

2 5 1 4 5 3

1 3 2 3 2 1

The '3' in the 1st row and 3rd column has not a '1' nearby...

Delete1 2 1 3 4 1

ReplyDelete3 2 3 2 2 4

3 1 4 3 1 3

2 4 5 4 5 2

1 3 4 1 3 1

2 2 1 2 2 1

The '4' in the second last row, the '4' in between the two '5's and the '4' above one '5' do not have a '2' nearby..

ReplyDelete4 2 4 5 1 4

ReplyDelete1 3 1 3 2 3

6 4 5 2 1 5

2 5 6 3 4 6

1 3 1 5 2 3

4 2 3 4 1 4

2 in first row doesnot have a one nearby

DeleteAs posted on http://domino.research.ibm.com/comm/wwwr_ponder.nsf/solutions/December2012.html

ReplyDeleteA solution without 5 (sum = 90):

2 4 1 4 2 1

1 3 2 3 4 1

2 4 4 1 4 2

4 1 3 2 3 4

3 2 4 4 1 1

1 3 1 3 2 3

Optimal solution of 93:

1 3 2 3 1 3

2 5 1 4 5 2

3 4 1 2 3 1

1 2 3 5 4 1

3 5 4 1 2 3

2 1 2 3 4 1

Thanks