(Remark. Fractions with a 1 in the numerator are called unit fractions. A sum of distinct unit fractions is called an Egyptian fraction. 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).)

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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/x_1 + 1/x_2= 1 \quad {\mathrm or} \quad 1/x_1 + 1/x_2 + 1/x_3 = 1?

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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

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Hint 3

One thing we might notice about this example is that 1/2 = 1/3 + 1/6 and therefore two numbers with (relatively) large denominators can add to make a number with a smaller denominator. Can we work backwards to represent a number with a small denominator as the sum of two numbers with larger denominators?

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The Rest of the Solution

Yes, 1/2 = 1/3 + 1/6 is one example of 1/n = 1/(n+1) + 1/(n(n+1)) . This trick allows us to start with 1 = 1/1 = 1/2 + 1/(1(2)) and work our way up, increasing the number of fractions by 1 at each step. We just have to be careful that we don't have any duplicates. Here's one way to finish with 7 (or 6 or 8) fractions:

1 = 1/1

1 = 1/2 + 1/2 (illegal, there's no distinct representation with two items)

1 = 1/2 + 1/3 + 1/6 (replaced 1/2)

1 = 1/2 + 1/4 + 1/6 + 1/12 (replaced 1/3)

1 = 1/2 + 1/5 + 1/6 + 1/12 + 1/20 (replaced 1/4)

1 = 1/2 + 1/5 + 1/7 + 1/12 + 1/20 + 1/42 (replaced 1/6)

1 = 1/2 + 1/6 + 1/7 + 1/12 + 1/20 + 1/30 + 1/42 (replaced 1/5)

1 = 1/2 + 1/6 + 1/8 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 (replaced 1/7)

Here we always converted the first fraction possible. Is it always possible to convert a fraction and not have a collision with another fraction? Yes, because we could always convert the last (largest-denominator) fraction.

Euclid Numbers

In the above example, in order to keep the denominators in the final answer small, we chose to split the largest fraction (the one with the smallest denominator), as long as that didn't produce a collision with another fraction. What would happen if we went to the other extreme, and attempted to get the largest possible denominators? We could try using the same kind of split, but always splitting the smallest fraction (largest denominator). We would start off getting the representations

1 = 1/1

1 = 1/2 + 1/2

1 = 1/2 + 1/3 + 1/6

as before, but after that we would have

1 = 1/2 + 1/3 + 1/7 + 1/42

1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1806

1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + 1/3263442

1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + 1/3263443 + 1/10650056950806

1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + 1/3263443 + 1/10650056950807 + 1/113423713055421844361000442

These are impressive fractions; each denominator is about the square of the preceding denominator. It's hard to believe those big denominators would really cancel out to leave us a simple integer, but they do. These denominators are sometimes called the Euclid numbers by analogy with Euclid's proof that there are an infinity of prime numbers. Euclid numbers can be defined more formally using a recursion:

e_1 = 2; \quad e_{n+1} = e_1...e_n + 1

so that we have an identity for all n

1 = 1/e_1 + ... + 1/e_n + 1/(e_{n+1} - 1)

The Euclid numbers appear because of the particular splitting rule that we chose, and of course there are lots of other ways to form fractions that add to 1. Here's a challenge for you (this one is hard): Prove that the Euclid number solutions gives the very largest possible denominators, in the sense that if 1 is the sum of unit fractions

1 = 1/x_1 + ... + 1/x_n

then each x_k \le e_n.

The Erdös-Straus Conjecture

There are many unsolved problems concerning unit fractions. Probably the most famous is the Erdös-Straus conjecture , due to Paul Erdös and E. G. Straus. Their conjecture is that, for any positive integer n, there is a solution in positive integers of the equation

4/n = 1/x + 1/y + 1/z.

This article dealt with the case n = 4, which has the solution x = 2, y = 3, and z = 6. The Erdös-Straus conjecture has been verified for all n < 10^8 and for numbers n having many special forms, but the general problem is still unsolved.

References

David Eppstein provides a page of links about Egyptian Fractions.
Richard K. Guy, Unsolved Problems in Number Theory, 2nd edition. Springer-Verlag, 1994. Section D11 describes unsolved problems in Egyptian Fractions, including the Erdös-Straus conjecture, and gives references to further results.
Euclid numbers are defined in: Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics 2nd edition, Addison-Wesley, 1994, Chapter 4. The Euclid number challenge is their exercise 4.59 (you can cheat by peeking at their solution). They attribute this result to D. R. Curtiss and Paul Erdös.
Wikipedia has an article on Egyptian fractions.
Martin Gardner, Fractal Music, Hypercards, and More...Mathematical Recreations from Scientific American, W. H. Freeman, 1991. Chapter 7, "Egyptian Fractions", pp. 100-109, has much historical and other information.