Perfect Numbers — Find the Numbers That Equal Their Own Parts

Some numbers have a magical property: add up all the smaller numbers that divide them, and you get the number back. The Greeks called these 'perfect'. You will find one with a handful of counters.

Take 6 counters (beads, buttons or coins). Find every way to divide them into equal groups: one group of 6, or 2 groups of 3, or 3 groups of 2, or 6 groups of 1. The group-SIZES that work (the divisors smaller than 6) are 1, 2 and 3.

Add those divisors: 1 + 2 + 3 = 6. The parts add back up to the number itself -- 6 is perfect! Now try 28 with 28 counters: its divisors are 1, 2, 4, 7 and 14, and they add to 28. Try 10 or 12 and you will find they do NOT work, which is why perfect numbers are so rare.

What your counters lead to. (1) The perfect numbers are strikingly rare: 6, 28, 496, 8128, then none until 33,550,336. (2) Around 300 BC Euclid found a recipe: whenever 2-to-the-power-p minus 1 is prime, you can build a perfect number from it -- which links them to the famous 'Mersenne primes' that a worldwide computer project still hunts today. (3) Every perfect number ever found is EVEN. (4) After 2,300 years, two questions Euclid could have asked remain unanswered: are there infinitely many, and does an ODD perfect number exist? Nobody has ever found one -- or proved one cannot exist. You have just handled the front edge of an open problem in mathematics.