35 / ANTIQUITY TO THE RENAISSANCE

Zhu Shijie

c. 1260–1320 CE·Yuan China

Stack oranges in triangular layers and count each pile. Subtract neighbouring totals, then subtract again: the changes settle to one steady number.

Explore the mathematics ↓

Imagined portrait of Zhu Shijie — Imagined likeness

01 / PILES AND DIFFERENCES

Find what stays steady inside growth.

Build piles of 1 to 5 layers and record each total.

Subtract, and subtract again, until the differences stop changing. Then add back up the table to predict the next pile, and build it to check.

Layers:

Oranges at one fixed size · each layer is shown again, laid flat, beside the pile

THE IDEAEach difference is a real piece of the pile: a new layer, a new row, a single orange. Because the third difference stays the same, adding back up the table predicts the next total, and Zhu’s rule n(n + 1)(n + 2) ÷ 6 gives any total at once.

The person behind the ideas

Travelled widely as a mathematics teacher.
Published Introduction to Mathematical Studies in 1299.
Published Jade Mirror of the Four Unknowns in 1303.

More of the mathematics

Piles of piles

Zhu went further: 1 + 4 + 10 + 20 + … adds up the pile totals themselves, and he gave its rule too, n(n + 1)(n + 2)(n + 3) ÷ 24.

Recruiting soldiers

Using differences, Zhu found how many soldiers are recruited when each day’s intake is a cube, 27, 64, 125, … . After 15 days: 23,400.