Quest for the mathematically ideal shoe-lacing

The NYT reports on the quest for the mathematically perfect way to lace a pair of shoes:

A multitude of trendy shoe fashion possibilities remains to be discovered, Dr. Polster said. A shoe with two rows of six eyelets offers 43,200 different paths for a shoelace to pass through every eyelet, even with the added condition that each eyelet must contribute to the essential purpose of pulling the two halves of the shoe together. (More precisely, this condition says the shoelace is not allowed to pass in a straight line through three consecutive eyelets on the same flap; otherwise, the middle of the three eyelets does not actively help close the shoe.)

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(Thanks, Steve!)