1. MAGIC WHEELS
August 1, 2012 Tofique Fatehi
tofiquef@yahoo.com
http://tofique.fatehi.us
WHEEL 1
Circumference = 1 unit
Take a wheel that is one unit in circumference. Mark a point A on the circumference. If the wheel is
rolled on a flat surface for one full turn, from A to A, it will have moved one unit.
WHEEL 2
Circumference = 3 units
Take a wheel that is three units in circumference. Mark a point A on the circumference and another
point B which is circumferentially one unit away from A. If the wheel is rolled on a flat surface from
A to B along the shorter arc, it will have moved one unit, whereas if it is moved from B to A along
the longer arc, it will have moved two units. And if it moved a full turn, A to A, then it will have
moved three units.
WHEEL 3
Circumference = 7 units
This 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 units
Magic Wheels
© 2012 Tofique Fatehi
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2. WHEEL 4
Circumference = 13 units
Wheel 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 units
Wheel 4 has another alternative to arc lengths - instead of 1, 2, 6, and 4 units it is also possible to
have 1, 3, 2, and 7 units.
WHEEL 5
Circumference = 21 units
Wheel 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 6
Circumference = 31 units
Wheel 6 is a very big wheel with 6 arcs of 1, 2, 5, 4, 6, and 13 units. It can move any integer units
from 1 to 31.
Wheel 6 has 5 alternatives to divide the circumference.
1, 2, 5, 4, 6, and 13 units, as above
1, 2, 7, 4, 12, and 5 units
1, 3, 2, 7, 8 and 10 units
1, 3, 6, 2, 5 and 14 units
1, 7, 3, 2, 4 and 14 units
Magic Wheels
© 2012 Tofique Fatehi
Page 2

3. PUZZLE
It can be noticed that in all cases, the circumference = 1 – n + n2 units, where n is the number of arcs
in the wheel.
 Why is the circumference 1 – n + n2 units?
 Find Wheel 7. How many alternatives can it have?
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Magic Wheels
© 2012 Tofique Fatehi
Page 3