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• How to tie the shortest shoelace?

TANJONG KATONG GIRLS SCHOOL Cho Soo Min
(Secondary 2E7 / 3E8) Tay Wen Niang Sheryl
(Secondary 2E7 / 3E8) Kwok Wai Sze (Secondary 2E7
/ 3E9)
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Aim

1. To find various lacing methods that are commonly
   used by people
2. To find the shortest lacing method

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Literature Review

• Criss-cross, or the bow-tie? There were
  different opinions.
• It turns out, however, that shorter lacings are
  possible if the lace doesn't have to pass
  alternately through the eyelets on the left and
  right side of the shoe.

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Criteria

• We found out the minimum criteria for a
  safe-to-use method.
• The shoelace must go through all the holes of the
  shoe.
• The shoelace need to go from one side to the
  other at least for five times.

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• Before showing the lacing methods, here are some
  of the points to be taken note of
• Regard the thickness of the shoelace be 0 cm
• There will not be any twisting of the shoelace
• Consider the length used to tie the knot would be
  same for all the methods.

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• Let the distance between the holes be

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Methodology

1. Use algorithms to find the shortest lacing
   method.
2. Form a formula for each lacing methods and
   substitute numbers.
3. Create a shorter lacing method if possible

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a) Algorithms

• What is algorithm?
• An algorithm is a specific set of instructions
  for carrying out a procedure or solving a
  problem, usually with the requirement that the
  procedure terminate at some point.
• There are many types of algorithms, and some of
  them can be used to find the shortest route in a
  specified area.
• Finding shortest lacing method could be regarded
  as a limited version of salesman problem, where
  we have to find the shortest route (in this case,
  lacing method). Thus, we went on investigating
  different algorithms that could suit our project.

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a) Algorithms
Algorithms Properties
Prims Algorithm Find the minimum spanning tree of a network
Kruskals Algorithm Go through all vertices (not necessary all edges) Hamiltonian Route Including edges with least weights without forming loops
Route Inspection Algorithm Go through all edges Eulerian Route Consider minimum retracing by pairing vertices with odd number of edges
Dijkstras Algorithm Find the shortest route between any 2 vertices in a network
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Remarks

• After judging each of the algorithms, we realized
  that none of them suits our project, as they will
  produce a route (in this case, lacing method)
  that does not suit our criteria - not going from
  one side to the other at least for five times.

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b) Form a formula for each lacing methods and
substitute numbers.
First of all, we chose the model shoe so that we
can substitute in the numbers.
Figure 1.1 The model shoe
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Straight European Method
Length used 5h 2d 4(vh2 4v2
)(Pythagoras Theorem) 5 x 5 2 x 5.3852 4 x
6.4031 61.4 cm (3 s.f.)
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Lattice Lacing
Length used h 4v 6 (vh2 9v2 )(Pythagoras
Theorem) 1 x 5 4 x 2 6 x 7.8102 59.9 cm
(3 s.f.)
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Sawtooth Lacing
Length used 5h 2v 4(vh2 4v2 )(Pythagoras
Theorem) 5 x 5 2 x 2 4 x 6.4031 54.6 cm
(3s.f.)
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Which method uses the least amount of shoelace?
Bow-tie Roman lacing Straight Easy Method Straight (Bar) Crossing
38.5 cm 42.5 cm 45 cm 45 cm
Lock Lacing Sawtooth Lacing Criss-crossMethod Lattice Lacing
52.0 cm 54.6 cm 58.8 cm 59.9 cm
Double Back Lacing Straight European Method Shoe Shop Lacing Train track lacing
60.2 cm 61.4 cm 63.1 cm 65 cm
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Substituting different measurementsLet h 4
cm, v 1.5 cm, d v42 1.52(Pythagoras)
4.2720 cm (5 s.f.)
Bow-tie Roman lacing Straight easy method Straight (bar) crossing
30.1 cm 33.1 cm 35 cm 35 cm
Lock lacing Sawtooth lacing Lattice method Criss-cross lacing
41.2 cm 43 cm 46.1 cm 46.7 cm
Double back lacing Straight European method Shoe shop lacing Train tracking
47 cm 48.5 cm 49.9 cm 51 cm
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Substituting different measurementsLet h 6
cm, v 2 cm, d v62 22(Pythagoras) 6.3246
cm (5 s.f.)
Bow-tie Roman lacing Straight easy method Straight (bar) crossing
43.3 cm 47.3 cm 50 cm 50 cm
Lock lacing Sawtooth lacing Lattice method Criss-cross lacing
60.6 cm 62.8 cm 64.9 cm 69.2 cm
Double back lacing Straight European method Shoe shop lacing Train tracking
67.7 cm 71.5 cm 73.3 cm 74 cm
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Substituting different measurementsLet h 5
cm, v 1.5 cm, d v52 1.52(Pythagoras)
5.2202 cm (5 s.f.)
Bow-tie Roman lacing Straight easy method Straight (bar) crossing
34.9 cm 37.9 cm 40 cm 40 cm
Lock lacing Sawtooth lacing Lattice method Double back lacing
49.8 cm 51.3 cm 51.4 cm 54.7 cm
Criss-cross lacing Straight European method Train track lacing Shoe shop lacing
57.2 cm 58.8 cm 60 cm 60.1 cm
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Substituting different measurementsLet h 5.5
cm, v 2 cm, d v5.52 22(Pythagoras)
5.8523 cm (5 s.f.)
Bow-tie Roman lacing Straight easy method Straight (bar) crossing
40.9 cm 44.9 cm 47.5 cm 47.5 cm
Lock lacing Sawtooth lacing Lattice method Double back lacing
56.3 cm 58.7 cm 62.3 cm 63.9 cm
Criss-cross lacing Straight European method Shoe shop lacing Train track lacing
64 cm 66.4 cm 68.2 cm 69.5 cm
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Substituting different measurementsLet h 4
cm, v 1 cm, d v42 12(Pythagoras) 4.1231
cm (5 s.f.)
Bow-tie Roman lacing Straight easy method Straight (bar) crossing
26.5 cm 28.5 cm 30 cm 30 cm
Lattice method Lock Lacing Sawtooth lacing Double back lacing
38 cm 39.0 cm 39.9 cm 41.8 cm
Criss-cross lacing Train track lacing Straight European method Shoe-shop method
45.2 cm 46.1 cm 46.1 cm 47 cm
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Result analysis

• For all our results, the bow-tie method was still
  the shortest method. Thus, it is safe to say that
  the bow-tie method is more or less the shortest
  lacing method.
• However, as some of the results changed, one
  should know the dimensions of the shoe before
  working out the formulae to find the shortest
  method as the result might vary.

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c) Create a shorter lacing method

• Is it possible to create a shorter lacing method?
• So far, we have found that the bow-tie method is
  the shortest lacing method.

Thus, we would have to first see if we Can
minimize the length used in the bow-tie Method.
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Is it possible to minimize the length used by
the bow-tie method?

• Properties of bow-tie method
• Bow-tie method goes through all the holes.
• It goes from one side to the other for exactly 5
  times without overlapping even once.

Bow-tie method
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Is it possible to minimize the length used by
the bow-tie method?
Figure 1. Bow-tie method
Figure 2. Shortened Bow-tie method
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Conclusion

• The length of the shoelace differs very much
  depending on which lacing method you are using,
  and the shoe shop method, which many shoe shops
  are using because it is regarded as the fastest
  way, actually is one of the longest lacing
  methods.

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The shortest lacing method is the bow-tie method
no matter what size of the shoe is.
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Recommendation

• one should use the bow-tie method if he / she
• has limited length of shoelace
• wants to shorten the shoelace to
• save the material

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dress with ribbons- joining two materials
together by punching holes and tying -
packaging and tying (help the factories in
packaging their products and cutting down on
their costs, especially since the current
Economic crisis.
Application
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Reference

1. Ian W. Fieggen (2003 2008)
2. Ivars Peterson (1999)
3. Burkard Polster (2006)
4. Burkard Polser (2002) Nature What is the best
   way to tie shoelace? (Dec 2002 Nature)
5. RANDY LEWIS (1986) Search for Perfect Shoelace
   Ties Him Up
6. HOLLIS W FIELD (1910) One Reason Why Men Go
   Insane Shoe Laces Made Extra Short.
7. Curet, William D. (2007)
8. Smith, Gregory S. (2007)
9. James M. Parks Solving Geometry Problems in
   Everyday Life
10. Wonder-how-to www.wonderhowto.com
11. Ivars Peterson (2002) Five-Suit Decks,
    Traffic-Jam Puzzles, and Other Treats