And, with a big spoiler warning, here are the answers (click the image for a larger version).
A sequence of puzzles (about sequences).
Lover of math. Bad at drawing.
Lover of math. Bad at drawing.
And, with a big spoiler warning, here are the answers (click the image for a larger version).
Such quirky, interesting explorations for kids! Inclusion of a slideshow in the form of Google Slides would be helpful to support teachers to try it out with kids. My thinking is something along the lines of a ‘slow reveal’ genre.
Thanks, good idea! I’ll put something together. (And I’ll brag here that my sister runs the Slow Reveal site!)
Is there a proof that the 46th Fibonacci number is less than 2 billion beyond just calculating? The nice simple proof does indeed show that the 47th number exceeds 2 billion, but the same argument also shows that 34 is the first Fibonacci number over 20.
Yeah, good question! Alas, it’s just a flaw in the puzzle — based on the information given, it totally could be the 46th rather than the 47th. (You can even make a puzzle out of selecting starting numbers other than 1, 1 which result in the 45th begin the first over 1 billion, and the 46th being the first over 2 billion.)
Or, to disambiguate, you can include the info that the 45th Fibonacci number is “just over 1.1 billion.” Then, if you assume the 46th number is over 2 billion, and work backwards to find the most recent terms, you’ll pretty quickly get a contradiction (44th = 46th – 45th, so must be over 800 million; similarly, 43rd must be under 400 million; but then the 42nd must be larger than the 43rd, which doesn’t make sense because the sequence is increasing by construction).
And after the string of factorials, there is Knuth’s arrow notation….! 😁😳 https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
One of my favorite mind-bending bits of mathematical notation!
“Finite” is still too big sometimes, IMO.
The answer sheet forgets to mention the word “eleven”, which is a possible ending for an odd number. Of course, it is not e-ban, either.
Ah, good point — the teens are omitted as well!
As long as you’re not trying to enumerate all of the t’s in the sentence there could be multiple solutions, such as “and twenty-fourth letter” or “and fifth from last” (Four words given for clarity.)
Hmm, and actually, those enumerations *could* be exhaustive! Aronson himself (setting aside my ambiguous paraphrasing) specified the increasing sequence, but there should be infinitely many sequences if you allow non-monotonicity, and I don’t even know how to think about the clever idea of “fifth from last”…
Hi Ben
Great 👍 We love such puzzles.
Thanks & Cheers
The Fab Four of Cley
🙂 🙂 🙂 🙂