The bishop on its diagonals: as far across as up

Typical question you must settle in under two seconds, eyes closed: a bishop on c4, does it attack f7? And g8? And a6? Or: list every square a bishop on d4 controls.

Why it is hard: a diagonal changes file AND rank at the same time. That is mechanically less obvious than a rook (one axis only). When several diagonals cross, the mind's eye gets lost, and the square colors that help you at the board are invisible blindfolded.

Good news: there is one exact mathematical rule that answers 100% of cases, plus a battery of shortcuts for speed. We teach them in that order.

A bishop NEVER leaves its color. That is the very first check, free and instant: if the target square is not the bishop's color, the answer is NO. No diagonal math needed. A light-squared bishop (say Bc4) will never attack d4, e5 or f6 (dark squares).

Reading the color must itself be instant: even sum of file + rank = dark, odd sum = light (f6 is 6+6 = 12, even, dark). If that is not automatic yet, go back to the square colors lesson first. Until color is instant, everything else stalls.

The whole board splits into two families of diagonals. This is the master technique, exact in 100% of cases.

Family "/" (rising, a1-h8 type): these diagonals climb to the right, file and rank increase together. Invariant: rank minus file is CONSTANT. On a1-h8 it is 0 everywhere (a1: 1-1=0, h8: 8-8=0). On a2-g8 it is +1, on b1-h7 it is -1.

Family "\" (falling, a8-h1 type): file increases while rank decreases. Invariant: file plus rank is CONSTANT. On a8-h1 the sum is 9 everywhere (a8: 1+8, h1: 8+1). On a7-g1 it is 8, on a6-f1 (the other diagonal of a c4 bishop) it is 7.

Instead of computing, you can trace the ray square by square from the bishop. Four directions, each step changes file and rank by +1 or -1:

• Up-right: file +1, rank +1 (c4, d5, e6, f7, g8)
• Up-left: file -1, rank +1 (c4, b5, a6)
• Down-right: file +1, rank -1 (c4, d3, e2, f1)
• Down-left: file -1, rank -1 (c4, b3, a2)

Walk until the edge or until a piece blocks the ray. A single piece on a diagonal stops that ray, and only that one: the bishop still sees its three other directions. This method is slower than the math rule, but it builds the mental image and it lists EVERY square (useful for "name everything the bishop controls"). Say the squares out loud at first.

The brain remembers named objects better than computations. Turn the recurring diagonals into named patterns:

• a1-h8 and h1-a8: the two longest (8 squares), the fianchetto highways (Bb2, Bg7, Bg2, Bb7). They pass between the four central squares without ever sharing one: a1-h8 goes through d4 and e5, h1-a8 through e4 and d5. Install this big X first.
• a2-g8: the diagonal of the attack on f7/g8 (Italian bishop on c4 or b3). Call it the "Italian diagonal".
• a6-f1: the other diagonal of the c4 bishop.
• c1-h6 and f1-a6: the classic exits of the dark-squared and light-squared bishops.

Landmark trick: hook each diagonal to a known reference. For example, every square whose file + rank sums to 7 sits on the c4 bishop's queenside diagonal (a6-f1); you only have to test the sum. And if you play an opening often, you memorize its diagonals NATURALLY: tie each diagonal to a position you already know rather than to an abstract number.

One consequence worth holding: on an empty board any two squares of the same color are two bishop moves apart at most, and squares of opposite colors are never connected at all. So a bishop is never far from where you want it, as long as the color is right.

Not all diagonals are equal: there are 15 per family, from length 1 to 8. A corner square (a1, h8) sits on one long diagonal and nothing else: its other diagonal is the corner square alone. A center square (d4, e4, d5, e5) controls the maximum, up to 13 squares. The closer to the edge, the less the bishop covers. Knowing this stops you searching for squares that do not exist (a bishop on h1 has nothing up-right, it is already the corner).

The board is symmetric: master the diagonals of the a1-d4 quadrant, then mirror to the other three. The visualization work divides by four.

Move the bishop anywhere, then test squares: the file gap and rank gap are computed in front of you, with the verdict.

Why bother? Because mastering diagonals unlocks the color complex: good bishop versus bad bishop (a bishop blocked by its own pawns), weak squares of one color around the king, open diagonals to grab. A player who "sees" diagonals instantly evaluates these themes effortlessly. Blindfold technique makes you better at the regular board too.

1. Instant color: call the color of random squares (parity). Absolute prerequisite.
2. The two diagonals of a square: for a given square, name its "/" and its "\".
3. Bishop slide: from a square, enumerate the ray in one direction.
4. YES/NO attack test: "does Bc4 attack X?" on a timer, applying "file gap = rank gap".
5. Bishop journeys: reach a target square by switching diagonals (the bishop often needs 2 legs).
6. Interleaving: alternate bishop, knight and memory drills rather than grinding one for days.

• Skipping the color filter. Computing a diagonal toward a square that is not even the right color. Filter by color first, always.
• Mixing the two families. "/" rises, so subtraction (difference); "\" falls, so addition (sum). Or simpler: "gap = gap", no signs.
• Missing the blocker. Blindfolded, it is easy to forget a piece cuts the diagonal. One piece stops the bishop.
• Computing everything. Recurring opening diagonals are pattern recognition, not arithmetic.