THE MAIN CHALLENGE:
The arrangements are: (1,3)(3,1)(2,4)(2,4)(3,5)(5,3)(4,6)(6,4)
The totals are: 36, 28, 52, 44, 68, 60, 84, 76 respectively
We noticed:
If the bigger number is in the blue and the smaller in red, the total will be 8 smaller than the other way around.
If you increase the dice numbers by one, the total increases 16 more.
This is because:
There are four less blue squares than red squares and the difference between the numbers is 2 so the total will be 8 smaller.
There are sixteen squares which are each one bigger.
We noticed:
That if you make the difference one larger, there are two less arrangements.
The difference to the total when you swap the numbers is larger when the difference is greater.
Increasing the difference will increase the total by 10 when the number in the blue square is the same. (e.g. (1,3) is 10 more than (1,2))
Increasing the difference will increase the total by 6 when the number in the red square is the same. (e.g. (3,1) is 6 more than (2,1))
You can't get 42 in less than a 2x2 grid or more than a 6x6 because:
The largest value in a 2x2 grid is 5 + 6 x 3 = 23
The smallest 6x6 grid is 15 x 2 + 21 x 1 = 51
You can't make 42 in a 5x5 grid because the only arrangements that are less than 42, (1,2) and (2,1) don't equal it.
In a 4x4 grid you can make 42 with (2,3) and in a 3x3 grid, you can use (2,6)