Is a "translation" just a parallel movement of the dots in a letter, so that a letter with only dots (2,1) and (3,2) would be a translation of "e," which has only (1,1) and (2,2)?
That's one example of translation. Translation is a rigid movement in any direction, so it could be horizontal, vertical, or diagonal (some combination of horizontal and vertical).
Spoilers for the "Making the rounds" puzzle below:
The number "6" is particularly nice: if you remove one chocolate from a given row, you have to remove at least 2 (to leave it even). Thus, you can only remove chocolates from up to 3 rows, so there is an untouched row. By symmetry, there is an untouched column. In the remaining rows and columns, we have to remove 2 each -- but this is the same as saying we have to leave exactly 1. So the answer is 4 times 4 times 3! = 96.
A semi-generalization with a nice classical answer is to ask how many ways you can take candies from an n-by-n box so that you leave exactly 2 in each row and column. But questions along these lines get hard very quickly ...
OK. I wasn't able to submit on time. But here's a generalized solution for an array of any size:
Σ(1,x)Σ(1,y)(2^xy*(1 - 2^(1-x)-2^(1-y)+2^(-2x)+2^(-2y)+2^(3-x-y)-2^(3-2x-y)-2^(3-x-2y)+2^(4-2x-2y))) + 1. Simple, right?
Are we counting no dots as one of the possible characters? I believe that's just used as a space. Otherwise there are only 63 possible characters...
I was including no dots as a character. But if folks submit an answer that's one less because they omitted this, I'll still mark as correct!
When translating a position, do the dots wrap to the other side when reaching an edge?
e.g. if so:
01
00
01
if moved one step up would become
00
01
01
I wouldn't consider those equivalent under translation. (In other words, the boundary conditions are not periodic.)
Nice puzzle
Is a "translation" just a parallel movement of the dots in a letter, so that a letter with only dots (2,1) and (3,2) would be a translation of "e," which has only (1,1) and (2,2)?
That's one example of translation. Translation is a rigid movement in any direction, so it could be horizontal, vertical, or diagonal (some combination of horizontal and vertical).
Thanks.
That's what I meant.
Spoilers for the "Making the rounds" puzzle below:
The number "6" is particularly nice: if you remove one chocolate from a given row, you have to remove at least 2 (to leave it even). Thus, you can only remove chocolates from up to 3 rows, so there is an untouched row. By symmetry, there is an untouched column. In the remaining rows and columns, we have to remove 2 each -- but this is the same as saying we have to leave exactly 1. So the answer is 4 times 4 times 3! = 96.
A semi-generalization with a nice classical answer is to ask how many ways you can take candies from an n-by-n box so that you leave exactly 2 in each row and column. But questions along these lines get hard very quickly ...
Dots with 6 removed=8008
Even with 6 removed=96
_ _ R R
_ R _ R
R _ _ R
R R R R
Now off to count how many there are.