Open Problems for Middle School

Eight unsolved problems that your students can understand, try by hand, and explore with technology. Nobody knows the answers, so when students gather evidence and make conjectures here, they are doing real mathematics.

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Primes · Grades 6–8Goldbach's ConjectureEvery even number bigger than 2 can be made by adding two primes.Primes · Grades 6–8Twin Prime ConjecturePrimes that are just 2 apart, like 11 and 13, keep showing up forever.Primes and squares · Grades 7–9Legendre's ConjectureBetween any two perfect squares that sit next to each other, there’s always a prime.Primes and subtraction · Grades 5–8Gilbreath's ConjectureList the primes. Subtract neighbors to make a new row. Keep going. Every new row starts with 1.Addition · Grades 4–7The 196 ProblemReverse a number and add. Repeat until it reads the same both ways. Does 196 ever get there?Fractions · Grades 5–8Erdős–Straus ConjectureAny fraction with 4 on top can be split into three fractions with 1 on top.Factors · Grades 5–8Odd Perfect NumbersA number is perfect when its factors add up to it, like 6 = 1 + 2 + 3. Is any perfect number odd?Rates and motion · Grades 7–10Lonely Runner ConjectureRunners start together on a round track, all at different speeds. Will every runner, at some moment, be far from everyone else?