Why is 3333360 the maximum score in Pac-Man?

Question

According to the Wikipedia page for Pac-Man, the highest possible score is 3333360 points. It's called a perfect game and it was achieved by 3 different people already.

What I want to know is: WHY is it exactly 3333360 points? It must be a technical limitation, but this number makes no sense in my opinion.

I know the game has 255 levels. 255 is 11111111 in binary and level 256 (that seems to exist mystically) brings the 8-bit digit to an overflow, which ends in a crash of the game.

But, 3333360 just seems so random.

Answer 1

The Pacman Museum has an article about getting 3,333,360 Points. And here's a video of some guy doing level 255 and 256, with important information regarding level 256.

Level 1 through 255

Eating dots: There are 240 regular dots per level, worth 10 points each, Netting 2400 points per level. Additionally, eating the four energizer dots, worth 50 points each, gives you another 200 points.
⇒ 255 × 2600 = 663000

Eating "fruit": There is one edible object per level that only appears for a certain amount of time, twice per level. Depending on the type of fruit, which in turn depends on the level you are playing, you get a different amount of points. In total, you get
⇒ 2 × (100 + 300 + 2 × 500 + 2 × 700 + 2 × 1000 + 2 × 2000 + 2 × 3000 + 243 × 5000) = 2459600

Eating Ghosts: Eating a ghost in the same energizer period gives you 200 points, and double that for every additional ghost you eat (400, 800 and 1600 for the 2nd through 4th ghost, respectively). You can do this 4 times per level, but only up to level 16, and on level 18, since the ghosts don't blink in level 17 and 19+ ⇒ 4 × 3000 × 17 = 204000

Adding these numbers yields 3326600, just 6760 points short of perfect.

Level 256

This odd number can be gotten in the glitched-out level 256.

Visible Dots: As you can see, a bit more than half of the screen is missing, which means you can only get 112 normal and 2 energizer dots.
⇒ 1220

Single Fruit: Due to the glitched screen, you can only get the key (the "fruit") once
⇒ 5000

Glitched Dots: There are 9 normal dots in the glitched out region that reappear every time you die. At 5 extra lives, you can get them 6 times
⇒ 6 × 90 = 540

Grand Total

(663000 + 1220 + 540 (dots)) + (2459600 + 5000 (fruit)) + (204000 (ghosts)) = 3333360

Answer 2

The score is limited because a glitch occurs on level 256 which overwrites half the screen with garbage. The game won't let a player advance from one board to the next without eating 244 dots and energizers, but the glitch overwrites many of the dots; this will leave the player unable to eat 244 dots and energizers, and thus unable to leave the level. In case you're wondering about why the glitch occurs, the machine code in Pac-Man to draw the fruits is similar to the C code:

unsigned char temp1, temp2;
unsigned char *ptr;

temp1 = level;
if (temp1 > 15) temp1 = 15;
temp2 = temp1;
if (temp2 > 7) temp2 = 7;
ptr = LOWER_RIGHT_ADDRESS;
do
{
  *ptr++ = shapes[temp1--];
} while(--temp);

Note that unlike many machines, Pac Man uses a rather curious screen memory layout which places consecutive bytes horizontally on the top and bottom of the screen, and vertically in the middle. This was most likely done to make it so that when drawing the main portion of the screen, memory addresses would be incremented by a power of two every eight scan lines (note that the "top" of the monitor is at the right side of the picture). Essentially, circuitry converted row/column indices to memory addresses with a mapping similar to:

//Using column values in the range 30 to 1, wrapping after 63...
address = (column & 32) ? 
          (row << 5) | (column & 31) :
          ((28 | (column & 3)) << 5)  | row);

but implemented in circuitry rather than code. This allowed screen addresses to be computed by using a pair of counters and a circuitry to select one of two permutations of those counters' bits. The hardware required to have memory address increase by 36 for each row rather than 32 would have been more complicated by comparison.