Hi, I try to answer your questions :
) but not the remainder of the board.
For the first one I didn't know th c4-f7 diagonal. What I did was extending b3 to a2. This diagonal will be one "lifted" from the a1-h8, so it will land on g8. Interpolating, f7 is indeed in that diagonal. I guess a crude way would be to always count the distance between the initial and final letters and numbers and if they match, they sit on the same diagonal.
The second one took me a bit of time in mind because I had to resort to visualisation. I had to get to either d6 or e5 and just had to decide on the shortest path using known knight jumps from my starting position. There were many ways to achieve this for both d6 and e5 in 3 moves. That was tough, I don't think I've had to visualise a board in my mind before.
1. Same as hicetnunc I know by experience c4 and f7 are connected and also that c4 and b3 are connected so no visualization I just know.
2. First I just tried moving it. d4 but then from here the first square the popped out for me was actually d6 (which I check was connected to f7) and that d4 to d6 is just 2 moves (because of color). I checked again to be sure but went backwards. f7 touches d6 and e5 ok I pick e5... e5 touches d3 and c4 ok I know those squares are both 2 moves from the starting square b3 (due to color).
3. No and no :) The central 16 squares always have color for me even if it's not relevant. Of the remaining outer 48 only the long diagonals come automatically (a8-b7, g7-h8, a1-b2, g2-h1). The others I usually have to think about for a second... if color isn't relevant for these squares then there's no color to me.
In game calculation is much move visual for me because there's a board in front of me. In these cases (if I'm not having a fuzzy headed day) I do literally see the calculated position on the board. If the position gets confused I go around and "erase" the pieces no longer on the board, while "adding" others to bring back the very clear image. This is only useful when you want to stop at a future position and look around for tactics i.e. calculate a line and go back to the future position to check another... sort of a save point. So during a fuzzy headed day it's more abstract (future pieces connected though lines of force and I check if those lines hit any weak/undefended pieces).
For these exercises I do not see a whole board. I use memory and visualize only pieces of the board at a time. e.g. memory c4 and f7 are connected / partial board visualization to check d6 and f7 are connected. (in my experience f7 is a very uncommon square for a knight, so couldn't just use memory).
In blindfold chess I do not see a whole board, it's much more abstract. The pieces interact with each other through logic and lines of force while the board is only seen in pieces as above when I checked d6 to f7 is a knight move. I've tried to create a whole board imagine during a blindfold game (just to try) and it's very hard for me and still mostly impossible (e.g. goal of focusing on any 4x4 segment and seeing the configuration of the pieces).
Which brings up a point you might find interesting. Seeing the entire 8x8 grid in the mind with the configuration of all the pieces seems useless to me. Even during a sighted exercise or game we only ever focus on relevant segments of the board. Maybe this is why unsighted 8x8 visualization is difficult for me (and I'd guess most people).
Really interesting answers so far. Thanks for posting everybody! Keep them coming.
wafflemaster, that is interesting. My experience is much like yours. I think sometimes seeing the entire board could be useful though. I'm thinking of certain positions where the winning tactic involves some counterintuitive sequence of moves all the way across the board (say a tactic that involves trapping the king on h8 where g8 is covered by a bishop and black has pawns on g7, g6, and white plays Qf6-a1-h1). Those kinds of long, backward moves are notoriously easy to overlook, probably because most of us do focus on the parts that seem relevant.
I've been trying to get myself to look at more of the board when doing tactics, because I've noticed that often my focus on particular parts of the board causes me to miss the solution, which I only see when I step back and look at every part of the board and then notice some minor clue in a different area of the board that I didn't initially think was relevant.
I guess even in those cases though, you don't have to see the entire 8x8 grid simultaneously. You could still focus on different parts, as long as you are able to transfer from one area of focus to another, so maybe the full 8x8 grid really isn't that useful.
I noticed that b3 and f7 were both on the a2-g8 diagonal, the phrase "a2-g8 diagonal" being a memorized and familiar phrase to me.
For the knight move...well to be perfectly honest, I'm quite out of practice with my blindfold. I'd work backward and forward from the endpoints to find paths that intersect. The relevant squares from f7 are e5, d6, g5, d8 (change either the letter or number by one, and the other by two is my mental process. eg. f-e is a change of 1, 7-5 is a change of 2, so f7 and e5 are knight moves apart.)
Sometimes the squares have color for me, but mostly not. That's one thing I need to work on yet is a better vision of the board. I find when I play that I remember structures, and can piece together a pretty full view of the board that way. Set up on your board pawns on f2, g3, h2, bishop on g2, king on g1, rook on f1, knight on f3 and take a look at it. Take it down, and an hour later try to recall and/or visualize all the pieces in their place. It'll be pretty easy, because that's a familiar structure, and "chunking" like that helps reduce the 'normal' parts of the board to something easy so you can remember the silly knight sitting on a8 for some reason. (for example)
When I visualize the board, I usually see all of it, but only maybe a 4x4 or 5x5 area is in focus...the other part is usually kind of foggy unless I consciously try to bring it into focus. I -can- get the whole board at once, if I try, but I agree with wafflemaster; it's usually a waste of effort, and it hinders calculation by trying to retain useless information.
1. Determine mentally (without looking at a board) if a bishop on b3 can move to f7 in one move.
Did you see the whole board in your mind, and then quickly move the bishop along the diagonal in your mind to see if f7 is on that diagonal? Or did you slowly move square by square (e.g., Bb3-c4-d5-e6-f7)? Or did you rely on memory that b3 and f7 are on the same diagonal? Or did you do it by some other means?
2. How would you determine mentally how many knight moves it takes to get from b3 to f7? Do you see the entire board as you're working, or just the area around the knight's current location?
3. When you visualize a piece on a square, does the square always have a color for you, or does the color not occur to you when it isn't relevant to the task?
I'm curious about how different people visualize while calculating, and how many see basically a physically accurate representation of the board in their mind (with all the squares always having a color) versus other more abstract ways of visualizing.