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Magic Wheels (1987)


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Original Magic Wheels article by Tofique Fatehi, as published in Science Today, in November 1987.

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Magic Wheels (1987)

  1. 1. LEivi_3G. o:ivilEslT l FUN WITH MATHS l M/ iClC WHEELS F a wheel, whose circumference is of unit length, is rolled along the round through one revolution, it wil? move fofward by a unit length. There is no magic in this. However, this is the simplest (and almost trivial) example of a magic wheel. We will defer the defini- tion of magic wheels till later. Consider next, a wheel whose cir- cumference is 3 units. The circumference is divided into two unequal segments, one of unit len th, and the other of two units (see Fig. al. This wheel will move c C= l3 b C=7 fig. I forward by l unit when rolled over on segment l, b 2 units when rolled over on segment , and b 3 units over a complete revolution. is is a magic wheel of the second order of magni- tude. _ _ Next, consider a wheel of the third order of ma nitude as shown in Fig. lb. This wheel as a circumference of 7 units, and is divided into three unequal segments. This wheel will move forward by l unit when rolled over on segment one, by two units for segment 2 and so on, and by 7 units when rolled over a complete revolution. Similarly, the wheel in Fig. lc, a wheel of the fourth order of magnitude, having a circumference of l3 units divided into four unequal se ments, can be made to move forward y any length‘ from l to 13 by ro ling it over one or more of its consecutive seg- menls. We may now attempt to define a magic wheel. A magic wheel of the nth order of magnitude is a wheel whose circumference is segmented into n un- equal parts of inle ral units of length in such a way that t e wheel will move forward by any rrriegral unit of length. ()8 1-‘. §I{‘rlif-V 'v'i'-. lI. £’l‘- 1‘--' inie’ ral unit of- U. FATEHI n=4, C= I2, andc<n(n- l)+7 Fig.2 froml to C, it rolled over on one or more of its consecutive segments, C being the circumference of the wheel. The next question is, how is C de- fined? This is easily answered it we take the sum of all the combinations of the n , segments, taking r consecutive seg- ments at a time, where r ranges from i to n. Thus there are n combinations for r= l, another n combinations for r=2, and so on till we have n combinations for r= n— l , and finally l combination for r= n, (that is, a full revolution) making a sum total of n(n—l)+l combinations. Hence a magic wheel of the nth order of ma nitude can have a maximum circum erence of C, ,., ,,, =n (n—l) + 1 units. All the wheels enumerated here magic wheels may be termed imperfect magic wheels. An impeded magic wheel must have all its individual seg- ments unequal, but some of its n(n—l)+l combinationsmay not be unique. The de ree of imperfection may be defined as t e len th required to be added to the circum erence to attain perfection. Fig.2 shows an imperfect magic wheel of fourth order where n=4,« C= 2 ' Fi . ‘3o shows a ma ic wheel of the , four1 order, which is iflerent from the ‘ wheel of Fig. lc, which is also of the fourth order, and both are perfect. Thus a rfect magic wheel of any given or er may not necessarily be unique. a C= l3 b C=27 Fig.3 Fi . 3b shows a fifth order perfect magic w eel, which incidentally is unique. Fig. 4 shows five different wheels of the sixth order of magnitude, and these are the only five possible perfect solu- tions lmirror-images are not counted). ll C=37 l 11 cr- Fig. 4 may be termed as perfect magic wheels. Perfect, because their respec- tive circumferences have a maximum value as given by the above formula ll l5 also possible to have magic wheels whose circumference is less than per- fect, and yet fit the definition. Such Constructing magic wheels is a chat- lenging, et absorbing pastime, and per apst e reader would like to under- take ihe challenge and attempt to con- struct a seventh order magic wheel. Be warned however, this is an addictive habit!