Each number in a grid represents how many other numbers can be seen from that square (as a queen in chess, horizontally, vertically, and diagonally, only as far as the first digit in that line.) Here is a grid using the numbers 1 thru 5:
3 
3 


5 
4 

1 




2 

2 

Meet these requirements for the following scenarios. Place at least one of each of the numbers 1 to N into
For example, if N=6, at least one each of 1,2,3,4,5,6 must be in the grid. Let N range from 1 to 8.
Source: Original.
Part #1  Smallest Area Rectangles 1 : 1 x 2 = 2 1 1 2 : 1 x 3 = 3 1 2 1 3 : 2 x 3 = 6 2 3 1 2 _ _ 4 : 3 x 3 = 9 2 3 _ 4 _ 2 1 _ _ 5 : 2 x 5 = 10 3 4 2 _ _ 3 5 _ _ 1 6 : 3 x 5 = 15 3 5 3 _ 1 5 6 _ _ _ 3 4 _ 2 _ 7 : 3 x 6 = 18 3 5 5 4 _ 1 5 7 6 4 _ _ 3 5 _ _ 2 _ 8 : 3 x 7 = 21 3 5 5 5 4 _ 1 5 8 7 6 4 _ _ 3 5 5 _ _ 2 _
Part #2  Minimize sum on an NxN square where there is at least one of each number 1 ... N 3 : 3 x 3 = 9, smallest sum = 8 2 2 _ 3 _ _ 1 _ _ 4 : 4 x 4 = 16, smallest sum = 12 2 3 _ _ 4 _ 2 _ 1 _ _ _ _ _ _ _ 5 : 5 x 5 = 25, smallest sum = 18 2 _ _ _ _ 3 5 _ _ 1 4 _ 2 _ _ 1 _ _ _ _ _ _ _ _ _ 6 : 6 x 6 = 36, smallest sum = 24 3 5 2 _ _ _ 4 6 _ _ _ _ 2 _ _ _ _ _ _ _ _ _ 1 _ _ 1 _ _ _ _ _ _ _ _ _ _Adrian matched Kirk's sums for 35 and also found these next two solutions.
7 : 7 x 7 = 49, smallest sum = 33 _ _ _ 1 _ _ _ _ 4 _ 6 _ 3 _ 2 _ _ _ _ _ _ _ 5 _ 7 _ 3 _ _ _ _ _ _ _ _ _ 2 _ _ _ _ _ _ _ _ 1 _ _ _ 8 : 8 x 8 = 64, smallest sum = 45 1 _ _ _ _ _ _ _ _ _ _ 3 _ 4 _ 3 _ _ _ _ _ _ _ _ _ _ 1 8 _ 7 _ _ _ _ _ _ _ _ 2 _ _ 2 _ 5 _ 6 _ _ _ _ _ _ 2 _ _ _ _ _ _ _ _ _ _ 2Joseph DeVincentis sent these solutions for part 3.
Part #3  Maximize sum on an NxN square where there is at least one of each number 1 ... N (with a maximum digit of N) 3 : largest sum = 8 2 2 _ 3 _ _ 1 _ _ 4 : largest sum = 20 _ 4 2 _ 3 4 _ _ 4 _ _ 2 1 _ _ _ 5 : largest sum = 40 2 _ _ _ _ 4 5 _ 4 2 _ 5 5 4 _ 3 _ 5 _ _ _ _ 1 _ _ 6 : largest sum = 82 2 _ _ _ _ _ 4 6 _ 5 4 3 5 6 6 6 6 _ _ 6 6 _ _ 4 3 _ 6 _ _ 3 _ _ 1 _ _ _Update 7/28/2006. Kirk found these solutions which have slightly higher sums.
4 : 4 x 4 = 16, sum = 24 3 4 3 _ 3 _ _ _ 4 _ 4 2 1 _ _ _ 5 : 5 x 5 = 25, sum = 50 3 4 5 _ 2 5 _ _ _ 4 5 _ _ 5 5 1 _ _ _ _ _ _ 4 4 3 6 : 6 x 6 = 36, sum = 88 3 _ 5 _ 5 3 4 _ _ 6 6 5 5 _ _ 6 6 _ 1 _ _ _ _ _ _ _ 6 6 6 4 _ 2 _ 5 4 _Kirk also solved some of part 3, allowing for any numbers to be in the NxN grid. I'd intended for the max number to be N, but I understand this interpretation, too.
Part #3  Maximize sum on an NxN square where there is at least one of each number 1 ... N 3 : 3 x 3 = 9, largest sum = 12 2 3 _ 4 _ 2 1 _ _ 4 : 4 x 4 = 16, largest sum = 34 3 4 _ _ 5 6 4 1 4 5 _ _ 2 _ _ _ 5 : 5 x 5 = 25, largest sum = 70 3 5 _ 2 _ 5 7 5 _ _ 5 8 6 _ 1 5 7 4 _ _ 3 4 _ _ _ 6 : 6 x 6 = 36, largest sum = 122 3 5 4 _ _ _ 5 8 7 4 _ _ 5 8 8 6 _ 1 5 8 8 5 _ _ 5 7 6 _ _ _ 3 5 _ 4 2 _