Sun Zi's Mathematical Classic

Mathematical Methods of the Counting Board

The method|original: 術; shù states: First, place 6,561 in the middle position to serve as the dividend|original: 實; shí.

Arrange 9 people below to serve as the divisor|original: 法; fǎ.

In the upper position, place 700. Use the upper 7 to “call out” (呼) refers to the oral recitation of multiplication tables used to determine how much to subtract from the dividend. call to the lower 9: “seven times nine is sixty-three.” Immediately subtract|original: 除; chú 6,300 from the middle position. This calculates $700 \times 9 = 6,300$. After subtracting this from 6,561, a remainder of 261 is left in the middle row.

Shift the lower position back by one place. This "shift" (退) aligns the divisor with the next decimal place (the tens) on the counting board. In the upper position, place 30. Use the upper 2 to call to the lower 9: “two times nine is eighteen.” There is a slight discrepancy here; the text specifies placing "30" but then uses the digit "2" for the calculation. Mathematically, 20 is the correct next digit for the quotient ($261 \div 9$). Immediately subtract 180 original: 二百八十 (280); the text reads 280, but based on the "call" of $9 \times 2$, this is likely a scribal error for 180 (一十八 with a trailing zero). from the middle position. Calculation: $261 - 180 = 81$.

Again, shift the lower position back by one more place. In the upper position, place 9. Use the upper 9 to call to the lower 9: “nine times nine is eighty-one.” Immediately subtract 81 from the middle position. The middle position is now completely exhausted.

Clear the lower position. The result obtained in the upper position is what each person receives. The final result in the upper row—the quotient—is 729.

From “eight times eight is sixty-four” down to “one times one is one,” all calculations follow this standard|original: 準此; zhǔn cǐ.