 |
|
|
|||||||||||||
 |
The Magnificent Seven
Submitted by Larry Engman, 12 December 1996. Original article by Valerio De Angelis, this article by Valerio De Angelis and Allen Stenger.
Find seven, positive, all different, whole numbers
x1, x2, x3, x4, x5, x6, x7
such that
1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5 + 1/x6 + 1/x7 = 1
Extra credit: What if we had asked for six such numbers? ...or eight?
(Remark. Fractions with a 1 in the numerator are called unit fractions, or less commonly Egyptian fractions or Ahmes fractions. The ancient Egyptians did not have a fully-developed concept of fractions; instead they performed most fractional calculations using a representation of the fraction as a sum of unit fractions as we are doing here. This method is described in detail in an ancient document today called the Rhind papyrus (after its owner) or the Ahmes papyrus (after its scribe).)
Hint 1
When you are faced with a math problem involving a ridiculously large number of things, such as seven fractions, practice on some smaller examples. For example: can you find two, or three, different positive whole numbers such that 1/x1 + 1/x2= 1 or 1/x1 + 1/x2 + 1/x3 = 1 ?
Hint 2
Two numbers turns out to be impossible, because they would have to be larger than 1, and distinct, so the smallest possibility would be 1/2 + 1/3 and this is already smaller than 1.
Three numbers does work, though, because if we pick the smallest possibilities 2 and 3 as the first two, we can complete it with 6:
1/2 + 1/3 + 1/6 = 1
Need another hint? Click here.
Click here for the complete solution.