Unit Fractions Puzzle

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Fraction Puzzle

Add a half, a third and a sixth, and you get one. That's the idea behind this puzzle - find a set of fractions that adds to one, each one over something, each different. 

Fractions like these - one over something - are called unit fractions. Get kids interested in this puzzle, and it can become great adding fractions practice. What's more, it links in to lots of other bits of fascinating math, such as perfect numbers.

A perfect number is one which equals the sum of all its factors. Six is a perfect number, because its factors are 1, 2 and 3. Divide 6=1+2+3 by six, and you get 1 = 1/6 + 1/3 + 1/2. Any perfect number gives a solution to this puzzle. Try it with the next two perfect numbers, 28 and 496.

On the other hand, if you have a number which is less than the sum of its factors, perhaps you can just pick some of its factors and add them up. The factors of 12 are 1, 2, 3, 4 and 6. Now, 2+4+6 makes 12, but dividing this by 12 still just gives 1 = 1/2 + 1/3 + 1/6. Oh, well, nobody said this method was perfect! Or can you see a different way to make 12 by adding up its factors? If you get stuck with 12, try some other numbers like it.

Perfect numbers are themselves linked to special prime numbers called Mersenne primes. The world's biggest known prime number is a Mersenne prime.

There are plenty of solutions to this puzzle that have nothing to do with perfect numbers. You could, for example, just start subtracting unit fractions from 1. Let's try that now.

  • If I take 1/3 from 1, I get 2/3.
  • If I take 1/4 from this, I get 5/12.
  • If I take 1/5 from this, I get 13/60
  • If I take 1/6 from this, I get 1/20.

So, I've found another solution to the puzzle - 1/3 + 1/4 + 1/5 + 1/6 + 1/20. Could I have done "better"?

Better might mean many different things - as few fractions as possible? With the smallest possible denominators? 

What you'll need

  • none
Variations:

Allow subtraction of fractions, instead of just addition. Then, make the target number 0 instead of 1.

How to play

  1. Find a collection of fractions, each one over something, each different, that adds up to 1.

  2. Give prizes for "good" solutions. These might be solutions with small denominators, or with as few fractions as possible. Or, solutions which don't use 1/2, or consider excluding 1/3 also. 

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