Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Magic Wheels (1987)

545 views

Published on

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

Published in: Education
  • Be the first to comment

  • Be the first to like this

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!

×