Just six different rings may be formed without breaking the conditions. Here is one way of effecting the arrangements.
| A | B | C | D | E | F | G | H | I | J | K | L | M |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | C | E | G | I | K | M | B | D | F | H | J | L |
| A | D | G | J | M | C | F | I | L | B | E | H | K |
| A | E | I | M | D | H | L | C | G | K | B | F | J |
| A | F | K | C | H | M | E | J | B | G | L | D | I |
| A | G | M | F | L | E | K | D | J | C | I | B | H |
Join the ends and you have the six rings.
Lucas devised a simple mechanical method for obtaining the n rings that may be formed under the conditions by 2n+1 children.