This group challenge is for students learning about prime numbers and prime factorization. Students will be working with fractions.
I have solved the problem by hand up to 30, and would enjoy to hear from anyone who pushes the envelope further 😉
PS. The #1 photo used in this puzzle is by Bri Cibene.
Nick Baxter has written a computer program to find the optimal results. Here is the output. He got up to 150 before his computer became sluggish. Let’s look at one number far beyond what I got up to: 50. Here it is:
(3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28) /
(2, 16, 26, 27, 30, 33, 34, 35, 36, 38, 40, 42, 44, 45, 46, 49, 50)
Equally interesting as the list of fractions is the sequence of the positive integers, n, that are improvements on n-1. Six (the first possible fraction) definitely offers an improvement on 5. Prime numbers never yield an improvement, but many composite numbers are also missing from the list. Improvements are never separated by too much.
Thank you Nick! That was such a beautiful set of fractions to receive 😉
Contact me: gord at mathpickle dot com, for any cool results you find.
Standards for Mathematical Practice
MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:
MP1 Toughen up!
This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.
MP3 Work together!
This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.
MP6 Be precise!
This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.
MP7 Be observant!
One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.