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Magic Wheels

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Magic Wheels

  1. 1. MAGIC WHEELSAugust 1, 2012 Tofique Fatehi tofiquef@yahoo.com http://tofique.fatehi.us WHEEL 1Circumference = 1 unitTake a wheel that is one unit in circumference. Mark a point A on the circumference. If the wheel isrolled on a flat surface for one full turn, from A to A, it will have moved one unit. WHEEL 2Circumference = 3 unitsTake a wheel that is three units in circumference. Mark a point A on the circumference and anotherpoint B which is circumferentially one unit away from A. If the wheel is rolled on a flat surface fromA to B along the shorter arc, it will have moved one unit, whereas if it is moved from B to A alongthe longer arc, it will have moved two units. And if it moved a full turn, A to A, then it will havemoved three units. WHEEL 3Circumference = 7 unitsThis wheel is 7 units in circumference, and has 3 points (A, B and C) marked on the circumference,dividing the circumference into 3 unequal arcs of 1 unit (AB), 2 units (BC) and 4 units (CA). A to B AB 1 unit B to C BC 2 units A to B to C ABC 3 units C to A CA 4 units C to A to B CAB 5 units B to C to A BC 6 units A to A ABCA 7 unitsMagic Wheels© 2012 Tofique Fatehi Page 1
  2. 2. WHEEL 4Circumference = 13 unitsWheel 4 is a bigger wheel that is divided into 4 arcs by 4 points (A, B, C, D). The arc lengths are 1, 2,6, and 4 units. You could move any number of units, from 1 through 13, like this: AB 1 unit BC 2 units ABC 3 units DA 4 units DAB 5 units CD 6 units DABC 7 units BCD 8 units ABCD 9 units CDA 10 units CDAB 11 units BCDA 12 units ABCDA 13 unitsWheel 4 has another alternative to arc lengths - instead of 1, 2, 6, and 4 units it is also possible tohave 1, 3, 2, and 7 units. WHEEL 5Circumference = 21 unitsWheel 5 has 5 Arcs of 1, 3, 10, 2, 5 units. You can use it to measure any integer units from 1 to 21. WHEEL 6Circumference = 31 unitsWheel 6 is a very big wheel with 6 arcs of 1, 2, 5, 4, 6, and 13 units. It can move any integer unitsfrom 1 to 31.Wheel 6 has 5 alternatives to divide the circumference.1, 2, 5, 4, 6, and 13 units, as above1, 2, 7, 4, 12, and 5 units1, 3, 2, 7, 8 and 10 units1, 3, 6, 2, 5 and 14 units1, 7, 3, 2, 4 and 14 unitsMagic Wheels© 2012 Tofique Fatehi Page 2
  3. 3. PUZZLEIt can be noticed that in all cases, the circumference = 1 – n + n2 units, where n is the number of arcsin the wheel.  Why is the circumference 1 – n + n2 units?  Find Wheel 7. How many alternatives can it have? Except as otherwise expressly permitted under copyright law, no copying, redistribution, retransmission, publication or commercial exploitation of this material will be permitted without the express permission of the copyright owner. In the event of any permitted copying, redistribution or publication of copyright material, no changes in or deletion of author attribution, trademark legend or copyright notice shall be made. You acknowledge that you do not acquire any ownership rights by downloading this copyrighted material.Magic Wheels© 2012 Tofique Fatehi Page 3

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