Page 2
Systematic Treatise on Arithmetic, Volume 14

Song of Decreasing Steps on a Journey

Three hundred and seventy-eight li to the pass,
A li is a traditional Chinese unit of distance, roughly equal to 500 meters or one-third of an English mile.
On the first day, his vigorous stride found no difficulty.
But the next day his legs ached, and his pace was halved,
Over six days, he traveled slowly to reach that pass.
If you wish to see the number of li traveled each day,
I ask you, sir, to calculate it carefully in return.

Answer:
On the first day, he traveled 192 li.
On the second day, he traveled 96 li.
On the third day, he traveled 48 li.
On the fourth day, he traveled 24 li.
On the fifth day, he traveled 12 li.
On the sixth day, he traveled 6 li.

Method: Set the total of 378 li as the dividend original: 實 (shí), the number to be divided. Arrange the proportional factors original: 衰 (shuāi), the relative weights assigned to each part of a distribution as 1, 2, 4, 8, 16, 32. Combining these gives 63 as the divisor original: 法 (fǎ), the number used to divide the dividend. Divide the dividend by the divisor to get 6 li, which is the distance for the sixth day. Progressively double this value for each preceding day to match the requirements of the question.

Song of Increasing Tiers of a Pagoda

This introduces a classic category of mathematical problems in ancient China involving geometric progressions, often visualized as a pagoda with increasing numbers of lights on each descending level.