Linked Rings in Antiquity
Although it hasn’t been determined when linked rings puzzles were invented, the concept of untangling linked rings has been embedded in Chinese culture at least as far back as the Warring States period (475–221 BCE), when philosopher Hui Shi (ca. 380–305 BCE) declared, “Linked rings can be separated.” Hui Shi’s explanation has been lost, but we know his concise paradox through the writings of others.
A Han dynasty (206 BCE–220 CE) history of the Warring States period contains a story that involves King Zheng of the Qin kingdom, the man who would later become Qin Shi Huang, the first emperor of China. King Zheng sent an emissary to present a set of jade linked rings to the Empress Dowager of the Qi Kingdom. The king’s message said, “The Qi people are quite intelligent, but can they untangle these rings?” The Empress showed the rings to her ministers, but none of them could untangle the rings. The Empress then took a hammer and broke the rings. She thanked the emissary from Qin and said, “Now they’re untangled!”
In the Ming dynasty (1368–1644) Yang Shen (1488–1559) wrote that the account of the Empress Dowager breaking the rings with a hammer was all wrong. “If this were true, she was simply a stupid woman. How could she outwit the Qin? The rings were an ingenious creation of jade craftsmen. There are two rings linked into one piece, but they can be untangled into two.” Then Yang Shen went on to say, “Nowadays, we also have an object called nine linked rings. It’s made of brass or iron instead of jade. It’s a toy for women and children.” This sixteenth century reference is the earliest known Chinese mention of the nine linked rings puzzle.
The earliest known Western description of a linked rings puzzle is by the Italian mathematician Luca Pacioli (1445–1517), who was a friend of Leonardo da Vinci. This appeared in Pacioli’s manuscript De Veribus Quantitatis, written around 1510. Pacioli states that “there can be three rings, or as many more as you want,” and he includes a solution for the case of seven rings. Pacioli’s description is only a few years earlier than poet Yang Shen’s mention of the puzzle, so this raises the question of whether linked rings puzzles originated in the West or in the East? Without further evidence, it’s impossible to say.
Nine Linked Rings Puzzles in the Palace
In the year 1713 of the Qing dynasty (1644–1911) Emperor Kangxi (reigned 1662–1722) was given a jade nine linked rings for his sixtieth birthday by the third daughter of his seventh son, Prince Chun.
Pu Yi (1906–1967), who assumed the throne in 1908 at the age of three to become the last emperor of the Qing dynasty, owned a beautiful silver nine linked rings puzzle with rings of jadeite.
Nine Linked Rings Puzzles in Literature, Music and Art
The nine linked rings puzzle is mentioned in the most popular novel in Chinese literature, Dream of the Red Chamber, written by Cao Xueqin (1715–1763) around 1760 and published in 1791. It contains a passage involving the two main characters in which Daiyu was in Baoyu’s room trying to untangle the nine linked rings with him.
The songbook Echos from White Snow, compiled by Hua Guangsheng in 1804, contains a song referring to the nine linked rings puzzle:
My lover gave me nine linked rings.
With my two hands I could not untangle them, I could not untangle them.
My lover, please untangle my nine linked rings, nine linked rings.
I will marry you and you will be my man.
The painter Yu Ji (1738–1823) was born in Hangzhou and gained some fame in Peking for his portraits of elegant ladies. In 1807 he painted a lady holding a nine linked rings puzzle. This portrait was purchased in Yangzhou in 1893 by the German sinologist Friedrich Hirth, who believed it to be a copy of one by the Ming dynasty master Tang Yin (1470–1523).
Around 1821 a writer who called himself Zhu Xiang Zhuren published six volumes of activities for girls and young ladies entitled Bits of Wisdom. He included an illustration of a nine linked rings puzzle and two charts that showed the recursive nature of the puzzle’s solution.
Wu Youru (1840–1893) was the leading illustrator of popular magazines portraying Chinese life in Shanghai during the last part of the nineteenth century. Wu Youru’s works reflected his keen observation of customs and manners. In 1892 he published a print entitled “Ingenious Undertaking of Rings” showing four young Shanghai ladies and a child in a luxurious setting. Two of the ladies are playing with nine linked rings puzzles, and the child appears to be reaching up for one of the puzzles.
Solving the Nine Linked Rings Puzzle
The beauty of many great puzzles is that their rules can be expressed very simply, yet the challenge of solving them requires much thought and patience. The nine linked rings is such a puzzle. The goal is to remove all nine rings that are trapped on the handle, and this can be accomplished in 341 steps by following two simple rules.
Before reading further, find a linked rings puzzle and see if you can solve it for yourself. Then analyze your solution and try to find an easy way to express it. Can you discover the two rules?
A procedure for solving the nine linked rings puzzle is based on the fact that at any given moment there are exactly two rings that can be taken off or put on the handle.
Rule A. Ring 1 can always be taken off or put on the handle.
Rule B. The only other ring that can be taken off or put on the handle is the ring immediately after the lead ring, where the lead ring is defined as the first ring that is on the handle.
If all the rings are on the handle, then there are two possibilities for the first move. (Following Rule A, the first ring could be taken off. Or following Rule B, the second ring could be taken off.) But after the first move, the only way to avoid backtracking is by alternating between applications of Rule A and Rule B.
To solve a linked rings puzzle with an odd number of rings, the first move must be taking off the first ring (Rule A). To solve a linked rings puzzle with an even number of rings, the first move must be taking off the second ring (Rule B). To take a ring off of the handle, simply slide it off and drop it through the handle. To put a ring on, raise it up through the handle and then slide it on.
Now let’s use the two rules to solve a three linked rings puzzle. The number of rings is odd, so we begin by applying Rule A:
Move 1. Apply Rule A by taking off Ring 1. Then Ring 2 becomes the lead ring.
Move 2. Apply Rule B by taking off Ring 3, the ring after the lead ring.
Move 3. Apply Rule A by putting on Ring 1. Then Ring 1 becomes the lead ring.
Move 4. Apply Rule B by taking off Ring 2, the ring after the lead ring.
Move 5. Apply Rule A by taking off Ring 1. Now all the rings are off!
Follow exactly the same rules to solve a nine linked rings puzzle. Nine is an odd number, so the first step must be taking off Ring 1. Then keep moving forward, alternating between Rules B and A. Be careful not to accidentally reverse directions and begin undoing the steps you’ve already completed. Good luck!
Cao Xueqin. Hong lou meng (Dream of the Red Chamber). 1791.
Liu Xiang. Zhan’guo ce (Chronicles of the Warring States). Circa 264–250 BCE.
Joseph Needham. Science and Civilisation in China, vol. 3. Cambridge, 1959.
Luca Pacioli. De Viribus Quantitatis. Circa 1500.
Yang Shen (1488–1559). Sheng’an ji (Sheng’an Collection), vol. 68.
Yu Chong En. Qiao huan (Ingenious Rings). Shanghai, 1958.
Zhu Xiang Zhuren. Xiao hui ji (Bits of Wisdom). [1821?].