Several people really liked my previous puzzle, including some high-school students 🙂 Some of you have sent me photos of your hand-written proofs. This level of enthusiasm for mathematics is totally awesome. Here is a detailed proof with some beautiful diagrams by Chris Grossack.
Here is my next puzzle:
Let and and .
Let and and .
All the above variables are positive integers and and and
Prove or disprove the following claim:
Claim: There exists a rational number such that and are rational numbers.
If you want to take small baby steps towards a proof, start with the following special cases:
Special case 1: and
Special case 2: and and
Quick homework problem: Prove the Special case 2 using a proof of my previous puzzle.
Have fun solving.
Some mathematician has said pleasure lies not in discovering truth, but in seeking it 🙂
I am working on a geometry problem and it lead me to make the following conjecture:
Conjecture 1: Given any convex polygon (v1, v2, …. v_n) drawn on a plane with integer edge lengths, there is a point x inside the polygon such that the euclidean distances between the pairs (x, v1), (x, v2), … (x, v_n) are all rational numbers.
Theorem: Conjecture 1 is true for n = 3.
Simple Homework: Prove Conjecture 1 for an equilateral triangle.
Conjecture 1 seems like a very natural geometry problem. Is it well-known ? If you know any references, please leave a comment or email me.
I am very excited to announce ANONYMOUS accounts in the freebits iOS app using decentralized public key cryptography. Username, email and password are NOT required.
Citizen journalists, photo journalists and anyone anywhere in the world can use freebits to prove the authenticity of their photos, videos (captured using our iOS app) without ever revealing their real-world identity.
A poll of 1,762 non-satellite TV stations and 3,379 radio outlets reveals journalists were threatened, assaulted, and even arrested at alarming rates across the U.S. in 2020. More details are in this article.
What is freebits ?
Freebits is a radically new kind of social media. It has built-in support for decentralized identities, privacy-preserving anonymous identities, AI-powered news aggregation, tamper-proof photos, videos, credentials and tamper-proof on-chain storage of socio-economic data (eg: unemployment numbers, prices of consumer goods, CPI to prevent data manipulation by governments and other centralized entities).Freebits is algorithm-free i.e., your posts will not be shadow-banned or hidden or suppressed.
Our mission is to enable journalists, creators, influencers, educators, podcasters, artists, politicians, activists, citizen journalists, photo journalists, data journalists, global leaders and everyone to speak freely to everyone globally, connect with their followers and get tips without any fear of censorship or shadow banning or de-platforming.
Freebits version 2.0 supports privacy-preserving anonymous accounts using public key cryptography. Username, email and password are not required. Citizen journalists, photo journalists and anyone anywhere in the world can use freebits to prove the authenticity of their photos, videos (captured using our iOS app) without ever revealing their real-world identity. Every photo and video shot using freebits app is a cryptographically-signed, verifiable and tamper-proof digital asset. The provenance and the digital rights of these assets are securely tracked using our Blockchain.
The intention of releasing this paper is to persuade everyone with factual & technical evidence that there is a global problem that must be addressed immediately & our solution is a superior method of solving it.
Today we are moving freebits (a decentralized twitter) from “beta” stage to “stable release” stage. Anyone from anywhere can signup and start using all the features.
Join Freebits and get free Euler cryptocurrency, proportional to the number of your twitter followers.
Only for the first 10,000 signups.
No hidden games. No scams. No spammy emails. You create your decentralized ID and own your private keys. You own your Euler. We will never ask you to send your bitcoin or any other crypto. We will never ask you to share your private keys or seed phrases.