Abundance often overwhelms, but restriction reaps riches. That’s true in mathematics and science, where you can often understand the whole better by looking at only a part of it first — restriction reaps riches. Egyptian fractions are one example in maths. In ancient Egypt, you could have any kind of fraction you liked so long as it was a reciprocal like 1/2, 1/3, 1/4 or 1/5 (well, there were two exceptions: 2/3 and 3/4 were also allowed).

So when mathematicians speak of “Egyptian fractions”, they mean those fractions that can be represented as a sum of reciprocals. Egyptian fractions are restricted and that reaps riches. Here’s one example: how many ways can you add *n* distinct reciprocals to make 1? When *n* = 1, there’s one way to do it: 1/1. When *n* = 2, there’s no way to do it, because 1 – 1/2 = 1/2. Therefore the summed reciprocals aren’t distinct: 1/2 + 1/2 = 1. After that, 1 – 1/3 = 2/3, 1 – 1/4 = 3/4, and so on. By the modern meaning of “Egyptian fraction”, there’s no solution for *n* = 2.

However, when *n* = 3, there is a way to do it:

• 1/2 + 1/3 + 1/6 = 1

But that’s the only way. When *n* = 4, things get better:

• 1/2 + 1/4 + 1/6 + 1/12 = 1

• 1/2 + 1/3 + 1/10 + 1/15 = 1

• 1/2 + 1/3 + 1/9 + 1/18 = 1

• 1/2 + 1/4 + 1/5 + 1/20 = 1

• 1/2 + 1/3 + 1/8 + 1/24 = 1

• 1/2 + 1/3 + 1/7 + 1/42 = 1

What about *n* = 5, *n* = 6 and so on? You can find the answer at the *Online Encyclopedia of Integer Sequences* (*OEIS*), where sequence A006585 is described as “Egyptian fractions: number of solutions to 1 = 1/x_{1} + … + 1/x_{n} in positive integers x_{1} < … < x_{n}”. The sequence is one of the shortest and strangest at the *OEIS*:

• 1, 0, 1, 6, 72, 2320, 245765, 151182379

When *n* = 1, there’s one solution: 1/1. When *n* = 2, there’s no solution, as I showed above. When *n* = 3, there’s one solution again. When *n* = 4, there are six solutions. And the *OEIS* tells you how many solutions there are for *n* = 5, 6, 7, 8. But *n* >= 9 remains unknown at the time of writing.

To understand the problem, consider the three reciprocals, 1/2, 1/3 and 1/5. How do you sum them? They have different denominators, 2, 3 and 5, so you have to create a new denominator, 30 = 2 * 3 * 5. Then you have to adjust the numerators (the numbers above the fraction bar) so that the new fractions have the same value as the old:

• 1/2 = 15/30 = (2*3*5 / 2) / 30

• 1/3 = 10/30 = (2*3*5 / 3) / 30

• 1/5 = 06/30 = (2*3*5 / 5) / 30

• 15/30 + 10/30 + 06/30 = (15+10+6) / 30 = 31/30 = 1 + 1/30

Those three reciprocals don’t sum to 1. Now try 1/2, 1/3 and 1/6:

• 1/2 = 18/36 = (2*3*6 / 2) / 36

• 1/3 = 12/36 = (2*3*6 / 3) / 36

• 1/6 = 06/36 = (2*3*6 / 6) / 36

• 18/36 + 12/36 + 06/36 = (18+12+6) / 36 = 36/36 = 1

So when *n* = 3, the problem consists of finding three reciprocals, 1/a, 1/b and 1/c, such that for a, b, and c:

• a*b*c = a*b + a*c + b*c

There is only one solution: a = 2, b = 3 and c = 6. When *n* = 4, the problem consists of finding four reciprocals, 1/a, 1/b, 1/c and 1/d, such that for a, b, c and d:

• a*b*c*d = a*b*c + a*b*d + a*c*d + b*c*d

For example:

• 2*4*6*12 = 576

• 2*4*6 + 2*4*12 + 2*6*12 + 4*6*12 = 48 + 96 + 144 + 288 = 576

• 2*4*6*12 = 2*4*6 + 2*4*12 + 2*6*12 + 4*6*12 = 576

Therefore:

• 1/2 + 1/4 + 1/6 + 1/12 = 1

When *n* = 5, the problem consists of finding five reciprocals, 1/a, 1/b, 1/c, 1/d and 1/e, such that for a, b, c, d and e:

• a*b*c*d*e = a*b*c*d + a*b*c*e + a*b*d*e + a*c*d*e + b*c*d*e

There are 72 solutions and here they are:

• 1/2 + 1/4 + 1/10 + 1/12 + 1/15 = 1 (#1)

• 1/2 + 1/4 + 1/9 + 1/12 + 1/18 = 1 (#2)

• 1/2 + 1/5 + 1/6 + 1/12 + 1/20 = 1 (#3)

• 1/3 + 1/4 + 1/5 + 1/6 + 1/20 = 1 (#4)

• 1/2 + 1/4 + 1/8 + 1/12 + 1/24 = 1 (#5)

• 1/2 + 1/3 + 1/12 + 1/21 + 1/28 = 1 (#6)

• 1/2 + 1/4 + 1/6 + 1/21 + 1/28 = 1 (#7)

• 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 1 (#8)

• 1/2 + 1/3 + 1/12 + 1/20 + 1/30 = 1 (#9)

• 1/2 + 1/4 + 1/6 + 1/20 + 1/30 = 1 (#10)

• 1/2 + 1/5 + 1/6 + 1/10 + 1/30 = 1 (#11)

• 1/2 + 1/3 + 1/11 + 1/22 + 1/33 = 1 (#12)

• 1/2 + 1/3 + 1/14 + 1/15 + 1/35 = 1 (#13)

• 1/2 + 1/3 + 1/12 + 1/18 + 1/36 = 1 (#14)

• 1/2 + 1/4 + 1/6 + 1/18 + 1/36 = 1 (#15)

• 1/2 + 1/3 + 1/10 + 1/24 + 1/40 = 1 (#16)

• 1/2 + 1/4 + 1/8 + 1/10 + 1/40 = 1 (#17)

• 1/2 + 1/4 + 1/7 + 1/12 + 1/42 = 1 (#18)

• 1/2 + 1/3 + 1/9 + 1/30 + 1/45 = 1 (#19)

• 1/2 + 1/4 + 1/5 + 1/36 + 1/45 = 1 (#20)

• 1/2 + 1/5 + 1/6 + 1/9 + 1/45 = 1 (#21)

• 1/2 + 1/3 + 1/12 + 1/16 + 1/48 = 1 (#22)

• 1/2 + 1/4 + 1/6 + 1/16 + 1/48 = 1 (#23)

• 1/2 + 1/3 + 1/9 + 1/27 + 1/54 = 1 (#24)

• 1/2 + 1/3 + 1/8 + 1/42 + 1/56 = 1 (#25)

• 1/2 + 1/3 + 1/8 + 1/40 + 1/60 = 1 (#26)

• 1/2 + 1/3 + 1/10 + 1/20 + 1/60 = 1 (#27)

• 1/2 + 1/3 + 1/12 + 1/15 + 1/60 = 1 (#28)

• 1/2 + 1/4 + 1/5 + 1/30 + 1/60 = 1 (#29)

• 1/2 + 1/4 + 1/6 + 1/15 + 1/60 = 1 (#30)

• 1/2 + 1/4 + 1/5 + 1/28 + 1/70 = 1 (#31)

• 1/2 + 1/3 + 1/8 + 1/36 + 1/72 = 1 (#32)

• 1/2 + 1/3 + 1/9 + 1/24 + 1/72 = 1 (#33)

• 1/2 + 1/4 + 1/8 + 1/9 + 1/72 = 1 (#34)

• 1/2 + 1/3 + 1/12 + 1/14 + 1/84 = 1 (#35)

• 1/2 + 1/4 + 1/6 + 1/14 + 1/84 = 1 (#36)

• 1/2 + 1/3 + 1/8 + 1/33 + 1/88 = 1 (#37)

• 1/2 + 1/3 + 1/10 + 1/18 + 1/90 = 1 (#38)

• 1/2 + 1/3 + 1/7 + 1/78 + 1/91 = 1 (#39)

• 1/2 + 1/3 + 1/8 + 1/32 + 1/96 = 1 (#40)

• 1/2 + 1/3 + 1/9 + 1/22 + 1/99 = 1 (#41)

• 1/2 + 1/4 + 1/5 + 1/25 + 1/100 = 1 (#42)

• 1/2 + 1/3 + 1/7 + 1/70 + 1/105 = 1 (#43)

• 1/2 + 1/3 + 1/11 + 1/15 + 1/110 = 1 (#44)

• 1/2 + 1/3 + 1/8 + 1/30 + 1/120 = 1 (#45)

• 1/2 + 1/4 + 1/5 + 1/24 + 1/120 = 1 (#46)

• 1/2 + 1/5 + 1/6 + 1/8 + 1/120 = 1 (#47)

• 1/2 + 1/3 + 1/7 + 1/63 + 1/126 = 1 (#48)

• 1/2 + 1/3 + 1/9 + 1/21 + 1/126 = 1 (#49)

• 1/2 + 1/3 + 1/7 + 1/60 + 1/140 = 1 (#50)

• 1/2 + 1/4 + 1/7 + 1/10 + 1/140 = 1 (#51)

• 1/2 + 1/3 + 1/12 + 1/13 + 1/156 = 1 (#52)

• 1/2 + 1/4 + 1/6 + 1/13 + 1/156 = 1 (#53)

• 1/2 + 1/3 + 1/7 + 1/56 + 1/168 = 1 (#54)

• 1/2 + 1/3 + 1/8 + 1/28 + 1/168 = 1 (#55)

• 1/2 + 1/3 + 1/9 + 1/20 + 1/180 = 1 (#56)

• 1/2 + 1/3 + 1/7 + 1/54 + 1/189 = 1 (#57)

• 1/2 + 1/3 + 1/8 + 1/27 + 1/216 = 1 (#58)

• 1/2 + 1/4 + 1/5 + 1/22 + 1/220 = 1 (#59)

• 1/2 + 1/3 + 1/11 + 1/14 + 1/231 = 1 (#60)

• 1/2 + 1/3 + 1/7 + 1/51 + 1/238 = 1 (#61)

• 1/2 + 1/3 + 1/10 + 1/16 + 1/240 = 1 (#62)

• 1/2 + 1/3 + 1/7 + 1/49 + 1/294 = 1 (#63)

• 1/2 + 1/3 + 1/8 + 1/26 + 1/312 = 1 (#64)

• 1/2 + 1/3 + 1/7 + 1/48 + 1/336 = 1 (#65)

• 1/2 + 1/3 + 1/9 + 1/19 + 1/342 = 1 (#66)

• 1/2 + 1/4 + 1/5 + 1/21 + 1/420 = 1 (#67)

• 1/2 + 1/3 + 1/7 + 1/46 + 1/483 = 1 (#68)

• 1/2 + 1/3 + 1/8 + 1/25 + 1/600 = 1 (#69)

• 1/2 + 1/3 + 1/7 + 1/45 + 1/630 = 1 (#70)

• 1/2 + 1/3 + 1/7 + 1/44 + 1/924 = 1 (#71)

• 1/2 + 1/3 + 1/7 + 1/43 + 1/1806 = 1 (#72)

All the sums start with 1/2 except for one:

• 1/2 + 1/5 + 1/6 + 1/12 + 1/20 = 1 (#3)

• 1/3 + 1/4 + 1/5 + 1/6 + 1/20 = 1 (#4)

Here are the solutions in another format:

(2,4,10,12,15), (2,4,9,12,18), (2,5,6,12,20), (3,4,5,6,20), (2,4,8,12,24), (2,3,12,21,28), (2,4,6,21,28), (2,4,7,14,28), (2,3,12,20,30), (2,4,6,20,30), (2,5,6,10,30), (2,3,11,22,33), (2,3,14,15,35), (2,3,12,18,36), (2,4,6,18,36), (2,3,10,24,40), (2,4,8,10,40), (2,4,7,12,42), (2,3,9,30,45), (2,4,5,36,45), (2,5,6,9,45), (2,3,12,16,48), (2,4,6,16,48), (2,3,9,27,54), (2,3,8,42,56), (2,3,8,40,60), (2,3,10,20,60), (2,3,12,15,60), (2,4,5,30,60), (2,4,6,15,60), (2,4,5,28,70), (2,3,8,36,72), (2,3,9,24,72), (2,4,8,9,72), (2,3,12,14,84), (2,4,6,14,84), (2,3,8,33,88), (2,3,10,18,90), (2,3,7,78,91), (2,3,8,32,96), (2,3,9,22,99), (2,4,5,25,100), (2,3,7,70,105), (2,3,11,15,110), (2,3,8,30,120), (2,4,5,24,120), (2,5,6,8,120), (2,3,7,63,126), (2,3,9,21,126), (2,3,7,60,140), (2,4,7,10,140), (2,3,12,13,156), (2,4,6,13,156), (2,3,7,56,168), (2,3,8,28,168), (2,3,9,20,180), (2,3,7,54,189), (2,3,8,27,216), (2,4,5,22,220), (2,3,11,14,231), (2,3,7,51,238), (2,3,10,16,240), (2,3,7,49,294), (2,3,8,26,312), (2,3,7,48,336), (2,3,9,19,342), (2,4,5,21,420), (2,3,7,46,483), (2,3,8,25,600), (2,3,7,45,630), (2,3,7,44,924), (2,3,7,43,1806)

**Note**

Strictly speaking, there are two solutions for *n* = 2 in genuine Egyptian fractions, because 1/3 + 2/3 = 1 and 1/4 + 3/4 = 1. As noted above, 2/3 and 3/4 were permitted as fractions in ancient Egypt.