The Math of Tying Shoes

This is inspired by each morning deciding on what order to put on socks and shoes, and when to tie my shoes.

Socks & Shoes

Each step: 2+4+4+2 = 12 choices
4 steps, 2 prerequisites = 6 orders

Socks & Tying Shoes

Each step: 2+4+6+6+4+2 = 24 choices
6 steps, 4 prerequisites = 20 orders

Pants, Socks, Tying Shoes

Each step: 3+3+5+6+6+4+2 = 29 choices
7 steps, 5 prerequisites = 52 orders

What is a general equation?

  Unsure.

  C(4,2) = 6
  C(6,3) = 20 = 4! - 2!
  C(6,4) = 15
  C(7,5) = 21

  I need to work on this more.

Socks & Shoes Slipped On

Here are all six possible ways to put on socks and slip-on shoes.

Left sock, left shoe, right sock, right shoe.
Left sock, right sock, left shoe, right shoe.
Left sock, right sock, right shoe, left shoe.
Right sock, left sock, left shoe, right shoe.
Right sock, left sock, right shoe, left shoe.
Right sock, right shoe, left sock, left shoe.

Socks & Shoes Tied

Here are all 20 ways you can put on socks and shoes and tie them.

Left sock, left shoe, left tied, right sock, right shoe, right tied.
Left sock, left shoe, right sock, left tied, right shoe, right tied.
Left sock, left shoe, right sock, right shoe, left tied, right tied.
Left sock, left shoe, right sock, right shoe, right tied, left tied.
Left sock, right sock, left shoe, left tied, right shoe, right tied.
Left sock, right sock, left shoe, right shoe, left tied, right tied.
Left sock, right sock, left shoe, right shoe, right tied, left tied.
Left sock, right sock, right shoe, left shoe, left tied, right tied.
Left sock, right sock, right shoe, left shoe, right tied, left tied.
Left sock, right sock, right shoe, right tied, left shoe, left tied.
Right sock, left sock, left shoe, left tied, right shoe, right tied.
Right sock, left sock, left shoe, right shoe, left tied, right tied.
Right sock, left sock, left shoe, right shoe, right tied, left tied.
Right sock, left sock, right shoe, left shoe, left tied, right tied.
Right sock, left sock, right shoe, left shoe, right tied, left tied.
Right sock, left sock, right shoe, right tied, left shoe, left tied.
Right sock, right shoe, left sock, left shoe, left tied, right tied.
Right sock, right shoe, left sock, left shoe, right tied, left tied.
Right sock, right shoe, left sock, right tied, left shoe, left tied.
Right sock, right shoe, right tied, left sock, left shoe, left tied.

Socks & Shoes Tied with Pants

Actually pants need to go on first, but they can be thrown into the mix for 145 different ways, enumerated here.

Pants, Left sock, left shoe, left tied, right sock, right shoe, right tied.
Pants, Left sock, left shoe, right sock, left tied, right shoe, right tied.
Pants, Left sock, left shoe, right sock, right shoe, left tied, right tied.
Pants, Left sock, left shoe, right sock, right shoe, right tied, left tied.
Pants, Left sock, right sock, left shoe, left tied, right shoe, right tied.
Pants, Left sock, right sock, left shoe, right shoe, left tied, right tied.
Pants, Left sock, right sock, left shoe, right shoe, right tied, left tied.
Pants, Left sock, right sock, right shoe, left shoe, left tied, right tied.
Pants, Left sock, right sock, right shoe, left shoe, right tied, left tied.
Pants, Left sock, right sock, right shoe, right tied, left shoe, left tied.
Pants, Right sock, left sock, left shoe, left tied, right shoe, right tied.
Pants, Right sock, left sock, left shoe, right shoe, left tied, right tied.
Pants, Right sock, left sock, left shoe, right shoe, right tied, left tied.
Pants, Right sock, left sock, right shoe, left shoe, left tied, right tied.
Pants, Right sock, left sock, right shoe, left shoe, right tied, left tied.
Pants, Right sock, left sock, right shoe, right tied, left shoe, left tied.
Pants, Right sock, right shoe, left sock, left shoe, left tied, right tied.
Pants, Right sock, right shoe, left sock, left shoe, right tied, left tied.
Pants, Right sock, right shoe, left sock, right tied, left shoe, left tied.
Pants, Right sock, right shoe, right tied, left sock, left shoe, left tied.
Left sock, pants, left shoe, left tied, right sock, right shoe, right tied.
Left sock, pants, left shoe, right sock, left tied, right shoe, right tied.
Left sock, pants, left shoe, right sock, right shoe, left tied, right tied.
Left sock, pants, left shoe, right sock, right shoe, right tied, left tied.
Left sock, pants, right sock, left shoe, left tied, right shoe, right tied.
Left sock, pants, right sock, left shoe, right shoe, left tied, right tied.
Left sock, pants, right sock, left shoe, right shoe, right tied, left tied.
Left sock, pants, right sock, right shoe, left shoe, left tied, right tied.
Left sock, pants, right sock, right shoe, left shoe, right tied, left tied.
Left sock, pants, right sock, right shoe, right tied, left shoe, left tied.
Right sock, pants, left sock, left shoe, left tied, right shoe, right tied.
Right sock, pants, left sock, left shoe, right shoe, left tied, right tied.
Right sock, pants, left sock, left shoe, right shoe, right tied, left tied.
Right sock, pants, left sock, right shoe, left shoe, left tied, right tied.
Right sock, pants, left sock, right shoe, left shoe, right tied, left tied.
Right sock, pants, left sock, right shoe, right tied, left shoe, left tied.
Right sock, pants, right shoe, left sock, left shoe, left tied, right tied.
Right sock, pants, right shoe, left sock, left shoe, right tied, left tied.
Right sock, pants, right shoe, left sock, right tied, left shoe, left tied.
Right sock, pants, right shoe, right tied, left sock, left shoe, left tied.
Left sock, right sock, pants, left shoe, left tied, right shoe, right tied.
Left sock, right sock, pants, left shoe, right shoe, left tied, right tied.
Left sock, right sock, pants, left shoe, right shoe, right tied, left tied.
Left sock, right sock, pants, right shoe, left shoe, left tied, right tied.
Left sock, right sock, pants, right shoe, left shoe, right tied, left tied.
Left sock, right sock, pants, right shoe, right tied, left shoe, left tied.
Right sock, left sock, pants, left shoe, left tied, right shoe, right tied.
Right sock, left sock, pants, left shoe, right shoe, left tied, right tied.
Right sock, left sock, pants, left shoe, right shoe, right tied, left tied.
Right sock, left sock, pants, right shoe, left shoe, left tied, right tied.
Right sock, left sock, pants, right shoe, left shoe, right tied, left tied.
Right sock, left sock, pants, right shoe, right tied, left shoe, left tied.

Created: 8 Oct 2018
Modified: 4 Nov 2019

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