Fibonacci’s Chinese Calendar

In a book completed in the year 1202, mathematician Leonardo of Pisa (also known as Fibonacci) posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on?

The total number of pairs, month by month, forms the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. Each new term is the sum of the previous two terms. This set of numbers is now called the Fibonacci sequence.

The Fibonacci numbers, F[x] (starting with 0), display a variety of patterns, including several interesting cycles. For example, the sequence begins with the numbers F[0] = 0, F[1] = 1, F[2] = 1, F[3] = 2, F[4] = 3, and F[5] = 5. The same numbers appear in the same order as the final digits of F[60] = 1,548,008,755,920; F[61] = 2,504,730,781,961; F[62] = 4,052,739,537,881; F[63] = 6,557,470,319,842; and so on. The same pattern holds for F[120] = 5,358,359,254,990,966,640,871,840; F[121] = 8,670,007,398,507,948,658,051,921; F[122] = 14,028,366,653,498,915,298,923,761; and so on. In other words, the final digits repeat every 60 values.

A cycle of 60 also plays an important role in the Chinese lunar calendar. The calendar uses two-character combinations to name each year. The first character represents one of the 10 “celestial stems,” and the second character represents one of the 12 “earthly branches.” The earthly branches constitute the Chinese zodiac of 12 animals: rat, ox, tiger, rabbit, dragon, snake, horse, ram, monkey, rooster, dog, and pig. According to the Chinese calendar, we’ve just entered the year of the snake.

The combination of a celestial stem cycle of 10 signs with a zodiac cycle of 12 animals generates 60 distinct year names (six cycles of stems and five cycles of branches). As a result, the years have names that are repeated every 60 years.

The curious coincidence of the Fibonacci cycle and the Chinese calendar cycle allowed Seok Sagong of Middlesex Community College in Middletown, Conn., to establish a one-to-one correspondence between the sequence of final digits of Fibonacci numbers and the names of years in the Chinese calendar. He described that relationship at last month’s Joint Mathematics Meetings in New Orleans.

Using Seok Sagong’s scheme, you can determine the year in the Chinese calendar that corresponds to any given Fibonacci number. Hence, the first year of the primary Chinese calendar cycle, jia-zi, corresponds to the first Fibonacci number, F[1]. The 60th year, gui-hai, corresponds to the 60th Fibonacci number, F[60].

You can also go the other way and determine the Fibonacci number that

corresponds to a given Chinese year.